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Sierra Nevada

The Sierra Nevada  is a mountain range in the Western United States, between the Central Valley of California and the Great Basin. The vast majority of the range lies in the state of California, although the Carson Range spur lies primarily in Nevada. The Sierra Nevada is part of the American Cordillera, an almost continuous chain of mountain ranges that forms the western "backbone" of the Americas.

The Sierra runs 400 mi north-south and its width ranges from 50 mi to 80 mi across east–west. Notable features include General Sherman, the largest tree in the world by volume; Lake Tahoe, the largest alpine lake in North America; Mount Whitney at 14505 ft, the highest point in the contiguous United States; and Yosemite Valley sculpted by glaciers from one-hundred-million-year-old granite, containing high waterfalls. The Sierra is home to three national parks, twenty-six wilderness areas, ten national forests, and two national monuments. These areas include Yosemite, Sequoia, and Kings Canyon National Parks; and Devils Postpile National Monument.

More than one hundred million years ago during the Nevadan orogeny, granite formed deep underground. The range started to uplift less than five million years ago, and erosion by glaciers exposed the granite and formed the light-colored mountains and cliffs that make up the range. The uplift caused a wide range of elevations and climates in the Sierra Nevada, which are reflected by the presence of five life zones (areas with similar plant and animal communities). Uplift continues due to faulting caused by tectonic forces, creating spectacular fault block escarpments along the eastern edge of the southern Sierra.

The Sierra Nevada has played an important role in the history of California and the United States. The California Gold Rush occurred in the western foothills from 1848 through 1855. Due to its inaccessibility, the range was not fully explored until 1912.

Name and etymology
Used in 1542 by Juan Rodríguez Cabrillo to describe a Pacific Coast Range (Santa Cruz Mountains), the term "Sierra Nevada" was a general identification of less familiar ranges toward the interior. In 1776, Pedro Font's map applied the name to the range currently known as the Sierra Nevada.

The literal translation is "snowy mountains", from sierra "a range of mountains", 1610s, from Spanish sierra "jagged mountain range", lit. "saw", from Latin serra "a saw"; and from fem. of Spanish nevado "snowy".

While many mountain ranges are unanimously referred to in the plural (Smokies, Rockies, Cascades, etc.), since Sierra is already pluralized in its native language, some locals who live in "the Sierra" are not hesitant to admonish those who refer to the area as "the Sierras". However, there are historical and literary references that use the plural, such as the 1871 collection of Joaquin Miller poems, Songs of the Sierras. Ansel Adams, in response to a publication of his photographs under the title Parmelian Prints of the High Sierras, commented, "The name Sierra is already a plural. To add an s is a linguistic, Californian, and mountaineering sin."

Geography
The Sierra Nevada lies primarily in central and eastern California, with the Carson Range, a small but historically important spur, extending into Nevada. West-to-east, the Sierra Nevada's elevation increases gradually from 500 ft in the Central Valley to more than 14000 ft atop the highest peaks of its crest 50 to 75 mi to the east. The east slope forms the steep Sierra Escarpment. Unlike its surroundings, the range receives a substantial amount of snowfall and precipitation due to orographic lift.

Setting
The Sierra Nevada's irregular northern boundary stretches from the Susan River and Fredonyer Pass to the North Fork Feather River. It represents where the granitic bedrock of the Sierra Nevada dives below the southern extent of Cenozoic igneous surface rock from the Cascade Range. It is bounded on the west by California's Central Valley, on the east by the Basin and Range Province, and on the southeast by the Mojave Desert. The southern boundary is at Tehachapi Pass.

Physiographically, the Sierra is a section of the Cascade–Sierra Mountains province, which in turn is part of the larger Pacific Mountain System physiographic division. The California Geological Survey states that "the northern Sierra boundary is marked where bedrock disappears under the Cenozoic volcanic cover of the Cascade Range."

Watersheds
The range is drained on its western slope by the Central Valley watershed, which discharges into the Pacific Ocean at San Francisco. The northern third of the western Sierra is part of the Sacramento River watershed (including the Feather, Yuba, and American River tributaries), and the middle third is drained by the San Joaquin River (including the Mokelumne, Stanislaus, Tuolumne, and Merced River tributaries). The southern third of the range is drained by the Kings, Kaweah, Tule, and Kern rivers, which flow into the endorheic basin of Tulare Lake, which rarely overflows into the San Joaquin during wet years.

The eastern slope watershed of the Sierra is much narrower; its rivers flow out into the endorheic Great Basin of eastern California and western Nevada. From north to south, the Susan River flows into intermittent Honey Lake, the Truckee River flows from Lake Tahoe into Pyramid Lake, the Carson River runs into Carson Sink, the Walker River into Walker Lake; Rush, Lee Vining and Mill Creeks flow into Mono Lake; and the Owens River into dry Owens Lake. Although none of the eastern rivers reach the sea, many of the streams from Mono Lake southwards are diverted into the Los Angeles Aqueduct which provides water to Southern California.

Elevation

The height of the mountains in the Sierra Nevada increases gradually from north to south. Between Fredonyer Pass and Lake Tahoe, the peaks range from 5000 ft to more than 9000 ft. The crest near Lake Tahoe is roughly 9000 ft high, with several peaks approaching the height of Freel Peak (10881 ft). Farther south, the highest peak in Yosemite National Park is Mount Lyell (13120 ft). The Sierra rises to almost 14000 ft with Mount Humphreys near Bishop, California. Finally, near Lone Pine, Mount Whitney is at 14505 ft, the highest point in the contiguous United States.

South of Mount Whitney, the elevation of the range quickly dwindles. The crest elevation is almost 10000 ft near Lake Isabella, but south of the lake, the peaks reach only a modest 8000 ft.

Notable features
There are several notable geographical features in the Sierra Nevada:

 * Lake Tahoe is a large, clear freshwater lake in the northern Sierra Nevada, with an elevation of 6225 ft and an area of 191 sqmi. Lake Tahoe lies between the main Sierra and the Carson Range, a spur of the Sierra.
 * Hetch Hetchy Valley, Yosemite Valley, Kings Canyon, and Kern Canyon are examples of many glacially-scoured canyons on the west side of the Sierra.
 * Yosemite National Park is filled with notable features such as waterfalls, granite domes, high mountains, lakes, and meadows.
 * Groves of giant sequoias Sequoiadendron giganteum occur along a narrow band of altitude on the western side of the Sierra Nevada. Giant sequoias are the largest trees in the world.
 * Two of the largest rivers in California, which form the Central Valley and drain into San Francisco Bay, derive most of their flow from the western slopes of the Sierra Nevada. The northern of the two is the Sacramento River (which also drains the adjacent Cascade Range and Klamath Range); the southern one is the San Joaquin River.

Communities
Communities in the Sierra Nevada include Paradise, South Lake Tahoe, Truckee, Grass Valley, Lee Vining, Mammoth Lakes, Sonora, Nevada City, Placerville, Pollock Pines, Portola, Auburn, Colfax and Kennedy Meadows.

Protected areas
Much of the Sierra Nevada consists of federal lands and is either protected from development or strictly managed. The mountain range is home to three National Parks – Yosemite, Kings Canyon, and Sequoia – and two national monuments – Devils Postpile and Giant Sequoia. Ten national forests span much of the mountain range's remaining area. Within these national parks, monuments, and forests lie 26 wilderness areas, which together protect 15.4% of the Sierra's 63118 km2 from logging and grazing.

The United States Forest Service and the Bureau of Land Management currently control 52% of the land in the Sierra Nevada. Logging and grazing are generally allowed on land controlled by these agencies, under federal regulations that balance recreation and development on the land.

The California Bighorn Sheep Zoological Area near Mount Williamson in the southern Sierra was established to protect the endangered Sierra Nevada bighorn sheep. Starting in 1981, hikers were unable to enter the Area from May 15 through December 15, in order to protect the sheep. As of 2010, the restriction has been lifted and access to the Area is open for the whole year.

Geologic history

The earliest rocks in the Sierra Nevada are metamorphic roof pendants of Paleozoic age, the oldest being metasedimentary rocks from the Cambrian in the Mount Morrison region. These dark-colored hornfels, slates, marbles, and schists are found in the western foothills (notably around Coarsegold, west of the Tehachapi Pass) and east of the Sierra Crest. The earliest granite of the Sierra started to form in the Triassic period. This granite is mostly found east of the crest and north of 37.2°N. In the Triassic and into the Jurassic, an island arc collided with the west coast of North America and raised a chain of volcanoes, in an event called the Nevadan orogeny. Nearly all subaerial Sierran Arc volcanoes have since disappeared; their remains were redeposited during the Great Valley Sequence and the subsequent Cenozoic filling of the Great Valley, which is the source of much of the sedimentary rock in California.

In the Cretaceous, a subduction zone formed at the edge of the continent. This means that an oceanic plate started to dive beneath the North American plate. Magma, formed through the subduction of the ancient Farallon Plate, rose in plumes (plutons) deep underground, their combined mass forming what is called the Sierra Nevada batholith. These plutons formed at various times, from 115 Ma to 87 Ma. The earlier plutons formed in the western half of the Sierra, while the later plutons formed in the eastern half of the Sierra. The Sierra Nevada formed the western margin of a high plateau to the east, the Nevadaplano.

Twenty million years ago, crustal extension associated with the Basin and Range Province caused extensive volcanism in the Sierra. About 10 Ma, the Sierra Nevada started to form when a block of crust between the Coast Range and the Basin and Range Province started to tilt to the west as heat from the Basin and Range extension thinned the eastern part of the block, making it more buoyant than the western portion of the block. Rivers started cutting deep canyons on both sides of the range. Lava filled some of these canyons, which have subsequently eroded leaving table mountains that follow the old river channels.

About 2.5 Ma, the Earth's climate cooled, and ice ages started. Glaciers carved out characteristic U-shaped canyons throughout the Sierra. The combination of river and glacier erosion exposed the uppermost portions of the plutons emplaced millions of years before, leaving only a remnant of metamorphic rock on top of some Sierra peaks.

Uplift of the Sierra Nevada continues today, especially along its eastern side. This uplift causes large earthquakes, such as the Lone Pine earthquake of 1872.

Climate and meteorology

The climate of the Sierra Nevada is influenced by the Mediterranean climate of California. During the fall, winter and spring, precipitation in the Sierra ranges from 20 to 80 in where it occurs mostly as snow above 6000 ft. Precipitation is highest on the central and northern portions of the western slope between 5000 and 8000 ft elevation, due to orographic lift. Above 8000 ft, precipitation diminishes on the western slope up to the crest, since most of the precipitation has been wrung out at lower elevations. Most parts of the range east of the crest are in a rain shadow, and receive less than 25 inches of precipitation per year. While most summer days are dry, afternoon thunderstorms are common, particularly during the North American Monsoon in mid and late summer. Some of these summer thunderstorms drop over an inch of rain in a short period, and the lightning can start fires. Summer high temperatures average 42 - 90 F. Winters are comparatively mild, and the temperature is usually only just low enough to sustain a heavy snowpack. For example, Tuolumne Meadows, at 8600 ft elevation, has winter daily highs about 40 F with daily lows about 10 F. The growing season lasts 20 to 230 days, strongly dependent on elevation. The highest elevations of the Sierra have an alpine climate.

The Sierra Nevada snowpack is the major source of water and a significant source of electric power generation in California. Many reservoirs were constructed in the canyons of the Sierra throughout the 20th century, Several major aqueducts serving both agriculture and urban areas distribute Sierra water throughout the state. However, the Sierra casts a rain shadow, which greatly affects the climate and ecology of the central Great Basin. This rain shadow is largely responsible for Nevada being the driest state in the United States.

Precipitation varies substantially from year to year. It is not uncommon for some years to receive precipitation totals far above or below normal.

The height of the range and the steepness of the Sierra Escarpment, particularly at the southern end of the range, produces a wind phenomenon known as the "Sierra Rotor". This is a horizontal rotation of the atmosphere just east of the crest of the Sierra, set in motion as an effect of strong westerly winds.

Because of the large number of airplanes that have crashed in the Sierra Nevada, primarily due to the complex weather and atmospheric conditions such as downdrafts and microbursts caused by geography there, a portion of the area, a triangle whose vertices are Reno, Nevada; Fresno, California; and Las Vegas, Nevada, has been dubbed the "Nevada Triangle", in reference to the Bermuda Triangle. Some counts put the number of crashes in the triangle at 2,000, including millionaire and record-breaking flyer Steve Fossett. Hypotheses that the crashes are related in some way to the United States Air Force's Area 51, or to the activities of extra-terrestrial aliens, have no evidence to support them.

Ecology

The Sierra Nevada is divided into a number of biotic zones, each of which is defined by its climate and supports a number of interdependent species. Life in the higher elevation zones adapted to colder weather, and to most of the precipitation falling as snow. The rain shadow of the Sierra causes the eastern slope to be warmer and drier: each life zone is higher in the east. A list of biotic zones, and corresponding elevations, is presented below:
 * The western foothill zone, 1000 - 2500 ft, with grassland, oak-grass savanna and chaparral-oak woodland. Gray pine (also known as Foothill pine) is intermixed with the oak woodland.
 * The Pinyon pine-Juniper woodland, 5000 - 7000 ft east side only.
 * The Sierra Nevada lower montane forest (indicator species: Ponderosa pine, Jeffrey pine), 2500 – west side, 7000 - 9000 ft east side. This biotic zone is notable for containing giant sequoia.
 * The Sierra Nevada upper montane forest (indicator species: Lodgepole pine, Red fir) 7000 – west side, 9000 – east side.
 * The Sierra Nevada subalpine zone (indicator species: Whitebark pine) 9000 – west side, 10500 – east side
 * The alpine region at greater than 10500 ft, and greater than 11500 ft east side.

Native Americans
Archaeological excavations placed Martis people of Paleo-Indians in northcentral Sierra Nevada during the period of 3,000 BCE to 500 CE. The earliest identified sustaining indigenous people in the Sierra Nevada were the Northern Paiute tribes on the east side, with the Mono tribe and Sierra Miwok tribe on the western side, and the Kawaiisu and Tübatulabal tribes in the southern Sierra. Today, some historic intertribal trade route trails over mountain passes are known artifact locations, such as Duck Pass with its obsidian arrowheads. The California and Sierra Native American tribes were predominantly peaceful, with occasional territorial disputes between the Paiute and Sierra Miwok tribes in the mountains. Washo and Maidu were also in this area prior to the era of European exploration and displacement.

Initial European-American exploration
American exploration of the mountain range started in 1827. Although prior to the 1820s there were Spanish missions, pueblos (towns), presidios (forts), and ranchos along the coast of California, no Spanish explorers visited the Sierra Nevada. The first Americans to visit the mountains were amongst a group led by fur trapper Jedediah Smith, crossing north of the Yosemite area in May 1827, at Ebbetts Pass.

In 1833, a subgroup of the Bonneville Expedition led by Joseph Reddeford Walker was sent westward to find an overland route to California. Eventually the party discovered a route along the Humboldt River across present-day Nevada, ascending the Sierra Nevada, starting near present-day Bridgeport and descending between the Tuolumne and Merced River drainage. The group may have been the first non-indigenous people to see Yosemite Valley. The Walker Party probably visited either the Tuolumne or Merced Groves of giant sequoia, becoming the first non-indigenous people to see the giant trees, but journals relating to the Walker party were destroyed in 1839, in a print shop fire in Philadelphia.

Starting in 1841, emigrants from the United States started to move to California via Sonora and Walker Passes.

In the winter of 1844, Lt. John C. Frémont, accompanied by Kit Carson, was the first European American to see Lake Tahoe. The Frémont party camped at 8050 ft.

Gold rush

The California Gold Rush began at Sutter's Mill, near Coloma, in the western foothills of the Sierra. On January 24, 1848, James W. Marshall, a foreman working for Sacramento pioneer John Sutter, found shiny metal in the tailrace of a lumber mill Marshall was building for Sutter on the American River. Rumors soon started to spread and were confirmed in March 1848 by San Francisco newspaper publisher and merchant Samuel Brannan. Brannan strode through the streets of San Francisco, holding aloft a vial of gold, shouting "Gold! Gold! Gold from the American River!"

On August 19, 1848, the New York Herald was the first major newspaper on the East Coast to report the discovery of gold. On December 5, 1848, President James Polk confirmed the discovery of gold in an address to Congress. Soon, waves of immigrants from around the world, later called the "forty-niners", invaded the Gold Country of California or "Mother Lode". Miners lived in tents, wood shanties, or deck cabins removed from abandoned ships. Wherever gold was discovered, hundreds of miners would collaborate to put up a camp and stake their claims.

Because the gold in the California gravel beds was so richly concentrated, the early forty-niners simply panned for gold in California's rivers and streams. However, panning cannot take place on a large scale, and miners and groups of miners graduated to more complex placer mining. Groups of prospectors would divert the water from an entire river into a sluice alongside the river, and then dig for gold in the newly exposed river bottom.

By 1853, most of the easily accessible gold had been collected, and attention turned to extracting gold from more difficult locations. Hydraulic mining was used on ancient gold-bearing gravel beds on hillsides and bluffs in the gold fields. In hydraulic mining, a high-pressure hose directed a powerful stream or jet of water at gold-bearing gravel beds. It is estimated that by the mid-1880s, 11 million troy ounces (340 metric tons) of gold (worth approximately US$16 billion in 2020 prices) had been recovered by "hydraulicking". A consequence of these extraction methods was that large amounts of gravel, silt, heavy metals, and other pollutants were washed into streams and rivers. , many areas still bear the scars of hydraulic mining, since the resulting exposed earth and downstream gravel deposits do not support plant life.

It is estimated that by 1855, at least 300,000 gold-seekers, merchants, and other immigrants had arrived in California from around the world. The huge numbers of newcomers brought by the Gold Rush drove Native Americans out of their traditional hunting, fishing and food-gathering areas. To protect their homes and livelihood, some Native Americans responded by attacking the miners, provoking counter-attacks on native villages. The Native Americans, out-gunned, were often slaughtered.

Thorough exploration
The Gold Rush populated the western foothills of the Sierra Nevada, but even by 1860, most of the Sierra was unexplored. The state legislature authorized the California Geological Survey to officially explore the Sierra (and survey the rest of the state). Josiah Whitney was appointed to head the survey. Men of the survey, including William H. Brewer, Charles F. Hoffmann and Clarence King, explored the backcountry of what would become Yosemite National Park in 1863. In 1864, they explored the area around Kings Canyon. In 1869, John Muir started his wanderings in the Sierra Nevada range, and in 1871, King was the first to climb Mount Langley, mistakenly believing he had summited Mount Whitney, the highest peak in the range. In 1873, Mount Whitney was climbed for the first time by 3 men from Lone Pine, California, on a fishing trip. From 1892 to 1897 Theodore Solomons made the first attempt to map a route along the crest of the Sierra.

Other people finished exploring and mapping the Sierra. Bolton Coit Brown explored the Kings River watershed in 1895–1899. Joseph N. LeConte mapped the area around Yosemite National Park and what would become Kings Canyon National Park. James S. Hutchinson, a noted mountaineer, climbed the Palisades (1904) and Mount Humphreys (1905). By 1912, the USGS published a set of maps of the Sierra Nevada, and the era of exploration was over.

Logging
Logging in the Sierra Nevada has significantly impacted the landscape. The logging industry in the Sierra Nevada started in the early 1800s, when settlers relied on hand tools and ox-teams. Before the California Gold Rush, the industry was relatively small, and most of the lumber used in the state was imported. However, as the demand for lumber to support the mining industry increased, logging became a major industry in the region.

Initially, most of the lumber produced in California was used in mining. The Comstock Lode was a major center for logging, with operations supplying lumber for the construction of mine structures, such as tunnels, shafts, and buildings, as well as fuel for the mines. Dan DeQuille observed in 1876, "the Comstock Lode may truthfully be said to be the tomb of the forests of the Sierra. Millions upon millions of feet of lumber are annually buried in the mines, nevermore to be resurrected.”

In the late 1800s, the logging industry moved westward due to the depletion of white pine forests in the upper Midwest. This shift was encouraged by the positive portrayal of the Sierra Nevada as a promising timber region. In 1859, Horace Greely marveled, "I never saw anything so much like good timber in the course of any seventy-five miles' travel as I saw in crossing the Sierra Nevada."

The logging industry experienced significant growth in the late 1800s due to several factors. The Timber and Stone Act of 1878 allowed individuals to claim ownership of old-growth timber tracts, which were later consolidated under joint-stock companies, such as those founded by Midwestern lumber magnates. These companies had the financial resources to transport timber from remote locations and build sawmills near the tracks of the Southern Pacific railroad which connected the San Joaquin Valley to the rest of the state in the 1870s. This facilitated the nationwide distribution of lumber. In addition, technological advancements, such as the shay locomotive and the v-shaped log flume, made it easier to transport lumber across mountainous terrain.

Conservation

The tourism potential of the Sierra Nevada was recognized early in the European history of the range. Yosemite Valley was first protected by the federal government in 1864. The Valley and Mariposa Grove were ceded to California in 1866 and turned into a state park. John Muir perceived overgrazing by sheep and logging of giant sequoia to be a problem in the Sierra. Muir successfully lobbied for the protection of the rest of Yosemite National Park: Congress created an Act to protect the park in 1890. The Valley and Mariposa Grove were added to the Park in 1906. In the same year, Sequoia National Park was formed to protect the Giant Sequoia: all logging of the Sequoia ceased at that time.

In 1903, the city of San Francisco proposed building a hydroelectric dam to flood Hetch Hetchy Valley. The city and the Sierra Club argued over the dam for 10 years, until the U.S. Congress passed the Raker Act in 1913 and allowed dam building to proceed. O'Shaughnessy Dam was completed in 1923.

Between 1912 and 1918, Congress debated three times to protect Lake Tahoe in a national park. None of these efforts succeeded, and after World War II, towns such as South Lake Tahoe grew around the shores of the lake. By 1980, the permanent population of the Lake Tahoe area grew to 50,000, while the summer population grew to 90,000. The development around Lake Tahoe affected the clarity of the lake water. In order to preserve the lake's clarity, construction in the Tahoe basin is currently regulated by the Tahoe Regional Planning Agency.

As the 20th century progressed, more of the Sierra became available for recreation; other forms of economic activity decreased. The John Muir Trail, a trail that followed the Sierra crest from Yosemite Valley to Mount Whitney, was funded in 1915 and finished in 1938. Kings Canyon National Park was formed in 1940 to protect the deep canyon of the Kings River.

In the 1920s, automobile clubs and nearby towns started to lobby for trans-Sierra highways over Piute Pass (which would have closed the gap in SR 168) and other locations. However, by end of the 1920s, the Forest Service and the Sierra Club decided that roadless wilderness in the Sierra was valuable, and fought the proposal. The Piute Pass proposal faded out by the early 1930s, with the Forest Service proposing a route over Minaret Summit in 1933. The Minaret Summit route was lobbied against by California's Governor Ronald Reagan in 1972. The expansion of the John Muir and Ansel Adams Wildernesses in the 1980s sealed off the Minaret Summit route.

A trans-Sierra route between Porterville and Lone Pine was proposed by local businessmen in 1923. Eventually, a circuitous route across the Sierra was built across Sherman Pass by 1976.

By 1964, the Wilderness Act protected portions of the Sierra as primitive areas where humans are simply temporary visitors. Gradually, 20 wilderness areas were established to protect scenic backcountry of the Sierra. These wilderness areas include the John Muir Wilderness (protecting the eastern slope of the Sierra and the area between Yosemite and Kings Canyon Parks), and wilderness within each of the National Parks.

The Sierra Nevada still faces a number of issues that threaten its conservation. Logging occurs on both private and public lands, including controversial clearcut methods and thinning logging on private and public lands. Grazing occurs on private lands as well as on National Forest lands, which include Wilderness areas. Overgrazing can alter hydrologic processes and vegetation composition, remove vegetation that serves as food and habitat for native species, and contribute to sedimentation and pollution in waterways. A recent increase in large wildfires like the Rim Fire in Yosemite National Park and the Stanislaus National Forest and the King Fire on the Eldorado National Forest, has prompted concerns. A 2015 study indicated that the increase in fire risk in California may be attributable to human-induced climate change. A study looking back over 8,000 years found that warmer climate periods experienced severe droughts and more stand-replacing fires and concluded that as climate is such a powerful influence on wildfires, trying to recreate presettlement forest structure may be difficult in a warmer future.

title: Sierra Nevada


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Differential calculus

In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.

The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point.

Differential calculus and integral calculus are connected by the fundamental theorem of calculus. This states that differentiation is the reverse process to integration.

Differentiation has applications in nearly all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of the velocity with respect to time is acceleration. The derivative of the momentum of a body with respect to time equals the force applied to the body; rearranging this derivative statement leads to the famous $F = ma$ equation associated with Newton's second law of motion. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories.

Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra.

Derivative
The derivative of $$f(x)$$ at the point $$x=a$$ is the slope of the tangent to $$(a,f(a))$$. In order to gain an intuition for this, one must first be familiar with finding the slope of a linear equation, written in the form $$y=mx+b$$. The slope of an equation is its steepness. It can be found by picking any two points and dividing the change in $$y$$ by the change in $$x$$, meaning that $$\text{slope } =\frac{\text{ change in }y}{\text{change in }x}$$. For, the graph of $$y=-2x+13$$ has a slope of $$-2$$, as shown in the diagram below:

 * $$\frac{\text{change in }y}{\text{change in }x}=\frac{-6}{+3}=-2$$

For brevity, $$\frac{\text{change in }y}{\text{change in }x}$$ is often written as $$\frac{\Delta y}{\Delta x}$$, with $$\Delta$$ being the Greek letter delta, meaning 'change in'. The slope of a linear equation is constant, meaning that the steepness is the same everywhere. However, many graphs such as $$y=x^2$$ vary in their steepness. This means that you can no longer pick any two arbitrary points and compute the slope. Instead, the slope of the graph can be computed by considering the tangent line—a line that 'just touches' a particular point. The slope of a curve at a particular point is equal to the slope of the tangent to that point. For example, $$y=x^2$$ has a slope of $$4$$ at $$x=2$$ because the slope of the tangent line to that point is equal to $$4$$:

The derivative of a function is then simply the slope of this tangent line. Even though the tangent line only touches a single point at the point of tangency, it can be approximated by a line that goes through two points. This is known as a secant line. If the two points that the secant line goes through are close together, then the secant line closely resembles the tangent line, and, as a result, its slope is also very similar:

The advantage of using a secant line is that its slope can be calculated directly. Consider the two points on the graph $$(x,f(x))$$ and $$(x+\Delta x,f(x+\Delta x))$$, where $$\Delta x$$ is a small number. As before, the slope of the line passing through these two points can be calculated with the formula $$\text{slope } = \frac{\Delta y}{\Delta x}$$. This gives

 * $$\text{slope} = \frac{f(x+\Delta x)-f(x)}{\Delta x}$$

As $$\Delta x$$ gets closer and closer to $$0$$, the slope of the secant line gets closer and closer to the slope of the tangent line. This is formally written as

 * $$\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$

The above expression means 'as $$\Delta x$$ gets closer and closer to 0, the slope of the secant line gets closer and closer to a certain value'. The value that is being approached is the derivative of $$f(x)$$; this can be written as $$f'(x)$$. If $$y=f(x)$$, the derivative can also be written as $$\frac{dy}{dx}$$, with $$d$$ representing an infinitesimal change. For example, $$dx$$ represents an infinitesimal change in x. In summary, if $$y=f(x)$$, then the derivative of $$f(x)$$ is

\frac{dy}{dx}=f'(x)=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x} $$

provided such a limit exists. We have thus succeeded in properly defining the derivative of a function, meaning that the 'slope of the tangent line' now has a precise mathematical meaning. Differentiating a function using the above definition is known as differentiation from first principles. Here is a proof, using differentiation from first principles, that the derivative of $$y=x^2$$ is $$2x$$:

\begin{align} \frac{dy}{dx}&=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x} \\ &= \lim_{\Delta x \to 0}\frac{(x+\Delta x)^2-x^2}{\Delta x} \\ &= \lim_{\Delta x \to 0}\frac{x^2+2x\Delta x+(\Delta x)^2-x^2}{\Delta x} \\ &= \lim_{\Delta x \to 0}\frac{2x\Delta x+(\Delta x)^2}{\Delta x} \\ &= \lim_{\Delta x \to 0}2x+\Delta x \\ \end{align} $$

As $$\Delta x$$ approaches $$0$$, $$2x+\Delta x$$ approaches $$2x$$. Therefore, $$\frac{dy}{dx}=2x$$. This proof can be generalised to show that $$\frac{d(ax^n)}{dx}=anx^{n-1}$$ if $$a$$ and $$n$$ are constants. This is known as the power rule. For example, $$\frac{d}{dx}(5x^4)=5(4)x^3=20x^3$$. However, many other functions cannot be differentiated as easily as polynomial functions, meaning that sometimes further techniques are needed to find the derivative of a function. These techniques include the chain rule, product rule, and quotient rule. Other functions cannot be differentiated at all, giving rise to the concept of differentiability.

A closely related concept to the derivative of a function is its differential. When $x$ and $y$ are real variables, the derivative of $f$ at $x$ is the slope of the tangent line to the graph of $f$ at $x$. Because the source and target of $f$ are one-dimensional, the derivative of $f$ is a real number. If $x$ and $y$ are vectors, then the best linear approximation to the graph of $f$ depends on how $f$ changes in several directions at once. Taking the best linear approximation in a single direction determines a partial derivative, which is usually denoted $∂y⁄∂x$. The linearization of $f$ in all directions at once is called the total derivative.

History of differentiation
The concept of a derivative in the sense of a tangent line is a very old one, familiar to ancient Greek mathematicians such as Euclid (c. 300 BC), Archimedes (c. 287–212 BC), and Apollonius of Perga (c. 262–190 BC). Archimedes also made use of indivisibles, although these were primarily used to study areas and volumes rather than derivatives and tangents (see The Method of Mechanical Theorems). The use of infinitesimals to compute rates of change was developed significantly by Bhāskara II (1114–1185); indeed, it has been argued that many of the key notions of differential calculus can be found in his work, such as "Rolle's theorem".

The mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), in his Treatise on Equations, established conditions for some cubic equations to have solutions, by finding the maxima of appropriate cubic polynomials. He obtained, for example, that the maximum (for positive $x$) of the cubic $ax^{2} – x^{3}$ occurs when $x = 2a / 3$, and concluded therefrom that the equation $ax^{2} = x^{3} + c$ has exactly one positive solution when $c = 4a^{3} / 27$, and two positive solutions whenever $0 < c < 4a^{3} / 27$. The historian of science, Roshdi Rashed, has argued that al-Tūsī must have used the derivative of the cubic to obtain this result. Rashed's conclusion has been contested by other scholars, however, who argue that he could have obtained the result by other methods which do not require the derivative of the function to be known.

The modern development of calculus is usually credited to Isaac Newton (1643–1727) and Gottfried Wilhelm Leibniz (1646–1716), who provided independent and unified approaches to differentiation and derivatives. The key insight, however, that earned them this credit, was the fundamental theorem of calculus relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes. For their ideas on derivatives, both Newton and Leibniz built on significant earlier work by mathematicians such as Pierre de Fermat (1607-1665), Isaac Barrow (1630–1677), René Descartes (1596–1650), Christiaan Huygens (1629–1695), Blaise Pascal (1623–1662) and John Wallis (1616–1703). Regarding Fermat's influence, Newton once wrote in a letter that "I had the hint of this method [of fluxions] from Fermat's way of drawing tangents, and by applying it to abstract equations, directly and invertedly, I made it general." Isaac Barrow is generally given credit for the early development of the derivative. Nevertheless, Newton and Leibniz remain key figures in the history of differentiation, not least because Newton was the first to apply differentiation to theoretical physics, while Leibniz systematically developed much of the notation still used today.

Since the 17th century many mathematicians have contributed to the theory of differentiation. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Augustin Louis Cauchy (1789–1857), Bernhard Riemann (1826–1866), and Karl Weierstrass (1815–1897). It was also during this period that the differentiation was generalized to Euclidean space and the complex plane.

The 20th century brought two major steps towards our present understanding and practice of derivation : Lebesgue integration, besides extending integral calculus to many more functions, clarified the relation between derivation and integration with the notion of absolute continuity. Later the theory of distributions (after Laurent Schwartz) extended derivation to generalized functions (e.g., the Dirac delta function previously introduced in Quantum Mechanics) and became fundamental to nowadays applied analysis especially by the use of weak solutions to partial differential equations.

Optimization
If $f$ is a differentiable function on $ℝ$ (or an open interval) and $x$ is a local maximum or a local minimum of $f$, then the derivative of $f$ at $x$ is zero. Points where $f ' (x) = 0$ are called critical points or stationary points (and the value of $f$ at $x$ is called a critical value). If $f$ is not assumed to be everywhere differentiable, then points at which it fails to be differentiable are also designated critical points.

If $f$ is twice differentiable, then conversely, a critical point $x$ of $f$ can be analysed by considering the second derivative of $f$ at $x$ : This is called the second derivative test. An alternative approach, called the first derivative test, involves considering the sign of the $x$ on each side of the critical point.
 * if it is positive, $x$ is a local minimum;
 * if it is negative, $x$ is a local maximum;
 * if it is zero, then $f(x) = x^{3}$ could be a local minimum, a local maximum, or neither. (For example, $x = 0$ has a critical point at $f(x) = ± x^{4}$, but it has neither a maximum nor a minimum there, whereas $x = 0$ has a critical point at $f '$ and a minimum and a maximum, respectively, there.)

Taking derivatives and solving for critical points is therefore often a simple way to find local minima or maxima, which can be useful in optimization. By the extreme value theorem, a continuous function on a closed interval must attain its minimum and maximum values at least once. If the function is differentiable, the minima and maxima can only occur at critical points or endpoints.

This also has applications in graph sketching: once the local minima and maxima of a differentiable function have been found, a rough plot of the graph can be obtained from the observation that it will be either increasing or decreasing between critical points.

In higher dimensions, a critical point of a scalar valued function is a point at which the gradient is zero. The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is called a "saddle point", and if none of these cases hold (i.e., some of the eigenvalues are zero) then the test is considered to be inconclusive.

Calculus of variations
One example of an optimization problem is: Find the shortest curve between two points on a surface, assuming that the curve must also lie on the surface. If the surface is a plane, then the shortest curve is a line. But if the surface is, for example, egg-shaped, then the shortest path is not immediately clear. These paths are called geodesics, and one of the most fundamental problems in the calculus of variations is finding geodesics. Another example is: Find the smallest area surface filling in a closed curve in space. This surface is called a minimal surface and it, too, can be found using the calculus of variations.

Physics
Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called differential equations. Physics is particularly concerned with the way quantities change and develop over time, and the concept of the "time derivative" — the rate of change over time — is essential for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:

 * velocity is the derivative (with respect to time) of an object's displacement (distance from the original position)
 * acceleration is the derivative (with respect to time) of an object's velocity, that is, the second derivative (with respect to time) of an object's position.

For example, if an object's position on a line is given by

 * $$x(t) = -16t^2 + 16t + 32, \,\!$$

then the object's velocity is

 * $$\dot x(t) = x'(t) = -32t + 16, \,\!$$

and the object's acceleration is

 * $$\ddot x(t) = x''(t) = -32, \,\!$$

which is constant.

Differential equations
A differential equation is a relation between a collection of functions and their derivatives. An ordinary differential equation is a differential equation that relates functions of one variable to their derivatives with respect to that variable. A partial differential equation is a differential equation that relates functions of more than one variable to their partial derivatives. Differential equations arise naturally in the physical sciences, in mathematical modelling, and within mathematics itself. For example, Newton's second law, which describes the relationship between acceleration and force, can be stated as the ordinary differential equation
 * $$F(t) = m\frac{d^2x}{dt^2}.$$

The heat equation in one space variable, which describes how heat diffuses through a straight rod, is the partial differential equation
 * $$\frac{\partial u}{\partial t} = \alpha\frac{\partial^2 u}{\partial x^2}.$$

Here $u(x,t)$ is the temperature of the rod at position $x$ and time $t$ and $α$ is a constant that depends on how fast heat diffuses through the rod.

Mean value theorem
The mean value theorem gives a relationship between values of the derivative and values of the original function. If $f(x)$ is a real-valued function and $a$ and $b$ are numbers with $a < b$, then the mean value theorem says that under mild hypotheses, the slope between the two points $(a, f(a))$ and $(b, f(b))$ is equal to the slope of the tangent line to $f$ at some point $c$ between $a$ and $b$. In other words,
 * $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

In practice, what the mean value theorem does is control a function in terms of its derivative. For instance, suppose that $f$ has derivative equal to zero at each point. This means that its tangent line is horizontal at every point, so the function should also be horizontal. The mean value theorem proves that this must be true: The slope between any two points on the graph of $f$ must equal the slope of one of the tangent lines of $f$. All of those slopes are zero, so any line from one point on the graph to another point will also have slope zero. But that says that the function does not move up or down, so it must be a horizontal line. More complicated conditions on the derivative lead to less precise but still highly useful information about the original function.

Taylor polynomials and Taylor series
The derivative gives the best possible linear approximation of a function at a given point, but this can be very different from the original function. One way of improving the approximation is to take a quadratic approximation. That is to say, the linearization of a real-valued function $f(x)$ at the point $x_{0}$ is a linear polynomial $a + b(x − x_{0})$, and it may be possible to get a better approximation by considering a quadratic polynomial $a + b(x − x_{0}) + c(x − x_{0})^{2}$. Still better might be a cubic polynomial $a + b(x − x_{0}) + c(x − x_{0})^{2} + d(x − x_{0})^{3}$, and this idea can be extended to arbitrarily high degree polynomials. For each one of these polynomials, there should be a best possible choice of coefficients $a$, $b$, $c$, and $d$ that makes the approximation as good as possible.

In the neighbourhood of $x_{0}$, for $a$ the best possible choice is always $f(x_{0})$, and for $b$ the best possible choice is always $f ' (x_{0})$. For $c$, $d$, and higher-degree coefficients, these coefficients are determined by higher derivatives of $f$. $c$ should always be $f (x''_{0})⁄2$, and $d$ should always be $f ' (x''_{0})⁄3!$. Using these coefficients gives the Taylor polynomial of $f$. The Taylor polynomial of degree $d$ is the polynomial of degree $d$ which best approximates $f$, and its coefficients can be found by a generalization of the above formulas. Taylor's theorem gives a precise bound on how good the approximation is. If $f$ is a polynomial of degree less than or equal to $d$, then the Taylor polynomial of degree $d$ equals $f$.

The limit of the Taylor polynomials is an infinite series called the Taylor series. The Taylor series is frequently a very good approximation to the original function. Functions which are equal to their Taylor series are called analytic functions. It is impossible for functions with discontinuities or sharp corners to be analytic; moreover, there exist smooth functions which are also not analytic.

Implicit function theorem
Some natural geometric shapes, such as circles, cannot be drawn as the graph of a function. For instance, if $f(x, y) = x^{2} + y^{2} − 1$, then the circle is the set of all pairs $(x, y)$ such that $f(x, y) = 0$. This set is called the zero set of $f$, and is not the same as the graph of $f$, which is a paraboloid. The implicit function theorem converts relations such as $f(x, y) = 0$ into functions. It states that if $f$ is continuously differentiable, then around most points, the zero set of $f$ looks like graphs of functions pasted together. The points where this is not true are determined by a condition on the derivative of $f$. The circle, for instance, can be pasted together from the graphs of the two functions $± √1 - x^{2}$. In a neighborhood of every point on the circle except and, one of these two functions has a graph that looks like the circle. (These two functions also happen to meet and, but this is not guaranteed by the implicit function theorem.)

The implicit function theorem is closely related to the inverse function theorem, which states when a function looks like graphs of invertible functions pasted together.

Other sources

 * Boman, Eugene, and Robert Rogers. Differential Calculus: From Practice to Theory. 2022, personal.psu.edu/ecb5/DiffCalc.pdf.
 * Boman, Eugene, and Robert Rogers. Differential Calculus: From Practice to Theory. 2022, personal.psu.edu/ecb5/DiffCalc.pdf.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Differential calculus


text:
Billie Holiday

Billie Holiday (born Eleanora Fagan; April 7, 1915 – July 17, 1959) was an American jazz and swing music singer. Nicknamed "Lady Day" by her friend and music partner, Lester Young, Holiday made a significant contribution to jazz music and pop singing. Her vocal style, strongly influenced by jazz instrumentalists, inspired a new way of manipulating phrasing and tempo. She was known for her vocal delivery and improvisational skills.

After a turbulent childhood, Holiday began singing in nightclubs in Harlem, where she was heard by producer John Hammond, who liked her voice. She signed a recording contract with Brunswick in 1935. Collaborations with Teddy Wilson produced the hit "What a Little Moonlight Can Do", which became a jazz standard. Throughout the 1930s and 1940s, Holiday had mainstream success on labels such as Columbia and Decca. By the late 1940s, however, she was beset with legal troubles and drug abuse. After a short prison sentence, she performed at a sold-out concert at Carnegie Hall. She was a successful concert performer throughout the 1950s with two further sold-out shows at Carnegie Hall. Because of personal struggles and an altered voice, her final recordings were met with mixed reaction but were mild commercial successes. Her final album, Lady in Satin, was released in 1958. Holiday died of heart failure on July 17, 1959, at age 44.

Holiday won four Grammy Awards, all of them posthumously, for Best Historical Album. She was inducted into the Grammy Hall of Fame and the National Rhythm & Blues Hall of Fame. In 2000, she was also inducted into the Rock & Roll Hall of Fame as an early influence; their website states that "Billie Holiday changed jazz forever". She was named one of the 50 Great Voices by NPR; and was ranked fourth on the Rolling Stone list of "200 Greatest Singers of All Time" (2023). Several films about her life have been released, most recently The United States vs. Billie Holiday (2021).

1915–1929: Childhood
Eleanora Fagan was born on April 7, 1915, in Philadelphia to African American unwed teenage couple Clarence Halliday and Sarah Julia "Sadie" Fagan (née Harris). Her mother moved to Philadelphia at age 19, after she was evicted from her parents' home in the Sandtown-Winchester neighborhood of Baltimore, Maryland, for becoming pregnant. With no support from her parents, she made arrangements with her older, married half-sister, Eva Miller, for Eleanora to stay with her in Baltimore. Not long after Eleanora was born, Clarence abandoned his family to pursue a career as a jazz banjo player and guitarist. Some historians have disputed Holiday's paternity, as a copy of her birth certificate in the Baltimore archives lists her father as "Frank DeViese". Other historians consider this an anomaly, probably inserted by a hospital or government worker. DeViese lived in Philadelphia, and Sadie, then known by her maiden name Harris, may have met him through her work. Harris married Philip Gough in 1920, but the marriage only lasted a few years.

Eleanora grew up in Baltimore and had a very difficult childhood. Her mother often took what were then known as "transportation jobs", serving on passenger railroads. Holiday was raised largely by Eva Miller's mother-in-law, Martha Miller, and suffered from her mother's absences and being in others' care for her first decade of life. Holiday's autobiography, Lady Sings the Blues, published in 1956, is inconsistent regarding details of her early life, but much was confirmed by Stuart Nicholson in his 1995 biography of the singer.

Holiday frequently skipped school which resulted in her being brought before the juvenile court at age nine. She was sent to the House of the Good Shepherd, a Catholic reform school for girls, where the nuns locked her in a room with a dead girl overnight as punishment for misbehavior. The experience traumatized her, and for years she would "dream about it and wake up hollering and screaming". After nine months, she was released on October 3, 1925, to her mother. Sadie had opened a restaurant, the East Side Grill, and mother and daughter worked long hours there. She dropped out of school at age 11.

On December 24, 1926, Sadie came home to discover a neighbor attempting to rape Holiday. She successfully fought back, and he was arrested. Officials sent Holiday to the House of the Good Shepherd under protective custody as a state witness in the rape case. Holiday was released in February 1927, when she was nearly 12. She found a job running errands in a brothel, and she scrubbed marble steps as well as kitchen and bathroom floors of neighborhood homes. Around this time, she first heard the records of Louis Armstrong and Bessie Smith. In particular, Holiday cited "West End Blues" as an intriguing influence, pointing specifically to the scat section duet with the clarinet as her favorite part. By the end of 1928, Holiday's mother moved to Harlem, New York, again leaving Eleanora with Martha Miller.

By early 1929, Holiday had joined her mother in Harlem.

1929–1935: Early career
As a young teenager, Holiday started singing in nightclubs in Harlem. She took her professional pseudonym from Billie Dove, an actress she admired, and Clarence Halliday, her probable father. At the outset of her career, she spelled her last name "Halliday", her father's birth surname, but eventually changed it to "Holiday", his performing name. The young singer teamed up with a neighbor, tenor saxophone player Kenneth Hollan. They were a team from 1929 to 1931, performing at clubs such as the Grey Dawn, Pod's and Jerry's on 133rd Street, and the Brooklyn Elks Club. Benny Goodman recalled hearing Holiday in 1931 at the Bright Spot. As her reputation grew, she played in many clubs, including the Mexico's and the Alhambra Bar and Grill, where she met Charles Linton, a vocalist who later worked with Chick Webb. It was also during this period that she connected with her father, who was playing in Fletcher Henderson's band.

Late in 1932, 17-year-old Holiday replaced the singer Monette Moore at Covan's, a club on West 132nd Street. Producer John Hammond, who loved Moore's singing and had come to hear her, first heard Holiday there in early 1933. Hammond arranged for Holiday to make her recording debut at age 18, in November 1933, with Benny Goodman. She recorded two songs: "Your Mother's Son-In-Law" and "Riffin' the Scotch", the latter being her first hit. "Son-in-Law" sold 300 copies, and "Riffin' the Scotch", released on November 11, sold 5,000 copies. Hammond was impressed by Holiday's singing style and said of her, "Her singing almost changed my music tastes and my musical life, because she was the first girl singer I'd come across who actually sang like an improvising jazz genius." Hammond compared Holiday favorably to Armstrong and said she had a good sense of lyric content at a young age.

In 1935, Holiday had a small role as a woman abused by her lover in Duke Ellington's musical short film Symphony in Black: A Rhapsody of Negro Life. She sang "Saddest Tale" in her scene.

1935–1938: Recordings with Teddy Wilson
In 1935, Holiday was signed to Brunswick by John Hammond to record pop tunes with pianist Teddy Wilson in the swing style for the growing jukebox trade. They were allowed to improvise on the material. Holiday's improvisation of melody to fit the emotion was highly skillful. Their first collaboration included "What a Little Moonlight Can Do" and "Miss Brown to You". "What a Little Moonlight Can Do" has been deemed her "claim to fame". Brunswick did not favor the recording session because producers wanted Holiday to sound more like Cleo Brown. However, after "What a Little Moonlight Can Do" was successful, the company began considering Holiday an artist in her own right. She began recording under her own name a year later for Vocalion in sessions produced by Hammond and Bernie Hanighen. Hammond said the Wilson-Holiday records from 1935 to 1938 were a great asset to Brunswick. According to Hammond, Brunswick was broke and unable to record many jazz tunes. Wilson, Holiday, Young, and other musicians came into the studio without written arrangements, reducing the recording cost. Brunswick paid Holiday a flat fee rather than royalties, which saved the company money. "I Cried for You" sold 15,000 copies, which Hammond called "a giant hit for Brunswick.... Most records that made money sold around three to four thousand."

Another frequent accompanist was tenor saxophonist Lester Young, who had been a boarder at her mother's house in 1934 and with whom Holiday had a rapport. Young said, "I think you can hear that on some of the old records, you know. Some time I'd sit down and listen to 'em myself, and it sound like two of the same voices ... or the same mind, or something like that." Young nicknamed her "Lady Day", and she called him "Prez".

1937–1938: Working for Count Basie and Artie Shaw
In late 1937, Holiday had a brief stint as a big-band vocalist with Count Basie. The traveling conditions of the band were often poor; they performed many one-nighters in clubs, moving from city to city with little stability. Holiday chose the songs she sang and had a hand in the arrangements, choosing to portray her developing persona of a woman unlucky in love. Her tunes included "I Must Have That Man", "Travelin' All Alone", "I Can't Get Started", and "Summertime", a hit for Holiday in 1936, originating in George Gershwin's Porgy and Bess the year before. Basie became used to Holiday's heavy involvement in the band. He said, "When she rehearsed with the band, it was really just a matter of getting her tunes like she wanted them, because she knew how she wanted to sound and you couldn't tell her what to do." Some of the songs Holiday performed with Basie were recorded. "I Can't Get Started", "They Can't Take That Away from Me", and "Swing It Brother Swing" are all commercially available. Holiday was unable to record in the studio with Basie, but she included many of his musicians in her recording sessions with Teddy Wilson.

Holiday found herself in direct competition with the popular singer Ella Fitzgerald. The two later became friends. Fitzgerald was the vocalist for the Chick Webb Band, which was in competition with the Basie band. On January 16, 1938, the same day that Benny Goodman performed his legendary Carnegie Hall jazz concert, the Basie and Webb bands had a battle at the Savoy Ballroom. Webb and Fitzgerald were declared winners by Metronome magazine, while DownBeat magazine pronounced Holiday and Basie the winners. Fitzgerald won a straw poll of the audience by a three-to-one margin.

By February 1938, Holiday was no longer singing for Basie. Various reasons have been given for why she was fired. Jimmy Rushing, Basie's male vocalist, called her unprofessional. According to All Music Guide, Holiday was fired for being "temperamental and unreliable". She complained of low pay and poor working conditions and may have refused to sing the songs requested of her or change her style. Holiday was hired by Artie Shaw a month after being fired from the Count Basie Band. This association placed her among the first black women to work with a white orchestra, an unusual arrangement at that time. This was also the first time a black female singer employed full-time toured the segregated U.S. South with a white bandleader. When Holiday faced racism, Shaw would often stick up for his vocalist. In her autobiography, Holiday describes an incident in which she was not permitted to sit on the bandstand with other vocalists because of racist policies. Shaw said to her, "I want you on the band stand like Helen Forrest, Tony Pastor and everyone else." When touring the South, Holiday would sometimes be heckled by members of the audience. In Louisville, Kentucky, a man called her a "nigger wench" and requested she sing another song. Holiday lost her temper and had to be escorted off the stage.

By March 1938, Shaw and Holiday had been broadcast on New York City's powerful radio station WABC (the original WABC, now WCBS). Because of their success, they were given an extra time slot to broadcast in April, which increased their exposure. The New York Amsterdam News reviewed the broadcasts and reported an improvement in Holiday's performance. Metronome reported that the addition of Holiday to Shaw's band put it in the "top brackets". Holiday could not sing as often during Shaw's shows as she could in Basie's; the repertoire was more instrumental, with fewer vocals. Shaw was also pressured to hire a white singer, Nita Bradley, with whom Holiday did not get along but had to share a bandstand. In May 1938, Shaw won band battles against Tommy Dorsey and Red Norvo, with the audience favoring Holiday. Although Shaw admired Holiday's singing in his band, saying she had a "remarkable ear" and a "remarkable sense of time", her tenure with the band was nearing an end. In November 1938, Holiday was asked to use the service elevator at the Lincoln Hotel in New York City, instead of the one used by hotel guests, because white patrons of the hotels complained. This may have been the last straw for her. She left the band shortly after. Holiday spoke about the incident weeks later, saying, "I was never allowed to visit the bar or the dining room as did other members of the band ... [and] I was made to leave and enter through the kitchen." There are no surviving live recordings of Holiday with Shaw's band. Because she was under contract to a different record label and possibly because of her race, Holiday was able to make only one record with Shaw, "Any Old Time". However, Shaw played clarinet on four songs she recorded in New York on July 10, 1936: "Did I Remember?", "No Regrets", "Summertime" and "Billie's Blues".

By the late 1930s, Holiday had toured with Count Basie and Artie Shaw, scored a string of radio and retail hits with Teddy Wilson, and became an established artist in the recording industry. Her songs "What a Little Moonlight Can Do" and "Easy Living" were imitated by singers across America and were quickly becoming jazz standards. In September 1938, Holiday's single "I'm Gonna Lock My Heart" ranked sixth as the most-played song that month. Her record label, Vocalion, listed the single as its fourth-best seller for the same month, and it peaked at number 2 on the pop charts, according to Joel Whitburn's Pop Memories: 1890–1954.

1939: "Strange Fruit" and Commodore Records
Holiday was in the middle of recording for Columbia in the late 1930s when she was introduced to "Strange Fruit", a song by Abel Meeropol based on his poem about lynching. Meeropol, a Jewish schoolteacher from the Bronx, used the pseudonym "Lewis Allan" for the poem, which was set to music and performed at teachers' union meetings. It was eventually heard by Barney Josephson, the proprietor of Café Society, an integrated nightclub in Greenwich Village, who introduced it to Holiday. She performed it at the club in 1939, with some trepidation, fearing possible retaliation. She later said that the imagery of the song reminded her of her father's death and that this played a role in her resistance to performing it.

For her performance of "Strange Fruit" at the Café Society, she had waiters silence the crowd when the song began. During the song's long introduction, the lights dimmed and all movement had to cease. As Holiday began singing, only a small spotlight illuminated her face. On the final note, all lights went out, and when they came back on, Holiday was gone. Holiday said her father, Clarence Holiday, was denied medical treatment for a fatal lung disorder because of racial prejudice, and that singing "Strange Fruit" reminded her of the incident. "It reminds me of how Pop died, but I have to keep singing it, not only because people ask for it, but because twenty years after Pop died the things that killed him are still happening in the South", she wrote in her autobiography. When Holiday's producers at Columbia found the subject matter too sensitive, Milt Gabler agreed to record it for his Commodore Records label on April 20, 1939. "Strange Fruit" remained in her repertoire for 20 years. She recorded it again for Verve. The Commodore release did not get any airplay, but the controversial song sold well, though Gabler attributed that mostly to the record's other side, "Fine and Mellow", which was a jukebox hit. "The version I recorded for Commodore", Holiday said of "Strange Fruit", "became my biggest-selling record". "Strange Fruit" was the equivalent of a top-twenty hit in the 1930s.

Holiday's popularity increased after "Strange Fruit". She received a mention in Time magazine. "I open Café Society as an unknown", Holiday said. "I left two years later as a star. I needed the prestige and publicity all right, but you can't pay rent with it." She soon demanded a raise from her manager, Joe Glaser. Holiday returned to Commodore in 1944, recording songs she made with Teddy Wilson in the 1930s, including "I Cover the Waterfront", "I'll Get By", and "He's Funny That Way". She also recorded new songs that were popular at the time, including, "My Old Flame", "How Am I to Know?", "I'm Yours", and "I'll Be Seeing You", a number one hit for Bing Crosby. She also recorded her version of "Embraceable You", which was inducted into the Grammy Hall of Fame in 2005.

1940–1947: Commercial success
Holiday's mother Sadie, nicknamed "The Duchess", opened a restaurant called Mom Holiday's. She used money from her daughter while playing dice with members of the Count Basie band, with whom she toured in the late 1930s. "It kept Mom busy and happy and stopped her from worrying and watching over me", Holiday said. Fagan began borrowing large amounts from Holiday to support the restaurant. Holiday obliged but soon fell on hard times herself. "I needed some money one night and I knew Mom was sure to have some", she said. "So I walked in the restaurant like a stockholder and asked. Mom turned me down flat. She wouldn't give me a cent." The two argued, and Holiday shouted angrily, "God bless the child that's got his own", and stormed out. With Arthur Herzog, Jr., a pianist, she wrote a song based on the lyric, "God Bless the Child", and added music. "God Bless the Child" became Holiday's most popular and most covered record. It reached number 25 on the charts in 1941 and was third in Billboard songs of the year, selling over a million records. In 1976, the song was added to the Grammy Hall of Fame. Herzog claimed Holiday contributed only a few lines to the lyrics. He said she came up with the line "God bless the child" from a dinner conversation the two had had.

On June 12, 1942, in Los Angeles, Holiday recorded "Trav'lin Light" with Paul Whiteman for a new label, Capitol Records. Because she was under contract to Columbia, she used the pseudonym "Lady Day". The song reached number 23 on the pop charts and number one on the R&B charts, then called the Harlem Hit Parade. On October 11, 1943, Life magazine wrote, "She has the most distinctive style of any popular vocalist, [and] is imitated by other vocalists."

Milt Gabler, in addition to owning Commodore Records, became an A&R man for Decca Records. He signed Holiday to Decca on August 7, 1944, when she was 29. Her first Decca recording was "Lover Man" (number 16 Pop, number 5 R&B), one of her biggest hits. The success and distribution of the song made Holiday a staple in the pop community, leading to solo concerts, rare for jazz singers in the late 1940s. Gabler said, "I made Billie a real pop singer. That was right in her. Billie loved those songs." Jimmy Davis and Roger "Ram" Ramirez, the song's writers, had tried to interest Holiday in the song. In 1943, a flamboyant male torch singer, Willie Dukes, began singing "Lover Man" on 52nd Street. Because of his success, Holiday added it to her shows. The record's flip side was "No More", one of her favorites. Holiday asked Gabler for strings on the recording. Such arrangements were associated with Frank Sinatra and Ella Fitzgerald. "I went on my knees to him", Holiday said. "I didn't want to do it with the ordinary six pieces. I begged Milt and told him I had to have strings behind me." On October 4, 1944, Holiday entered the studio to record "Lover Man", saw the string ensemble and walked out. The musical director, Toots Camarata, said Holiday was overwhelmed with joy. She may also have wanted strings to avoid comparisons between her commercially successful early work with Teddy Wilson and everything produced afterwards. Her 1930s recordings with Wilson used a small jazz combo; recordings for Decca often involved strings. A month later, in November, Holiday returned to Decca to record "That Ole Devil Called Love", "Big Stuff", and "Don't Explain". She wrote "Don't Explain" after she caught her husband, Jimmy Monroe, with lipstick on his collar.

Holiday did not make any more records until August 1945, when she recorded "Don't Explain" for a second time, changing the lyrics "I know you raise Cain" to "Just say you'll remain" and changing "You mixed with some dame" to "What is there to gain?" Other songs recorded were "Big Stuff", "What Is This Thing Called Love?", and "You Better Go Now". Ella Fitzgerald named "You Better Go Now" her favorite recording of Holiday's. "Big Stuff" and "Don't Explain" were recorded again but with additional strings and a viola. In 1946, Holiday recorded "Good Morning Heartache". Although the song failed to chart, she sang it in live performances; three live recordings are known.

In September 1946, Holiday began her only major film, New Orleans, in which she starred opposite Louis Armstrong and Woody Herman. Plagued by racism and McCarthyism, producer Jules Levey and script writer Herbert Biberman were pressed to lessen Holiday's and Armstrong's roles to avoid the impression that black people created jazz. The attempts failed because in 1947 Biberman was listed as one of the Hollywood Ten and sent to jail. Several scenes were deleted from the film. "They had taken miles of footage of music and scenes", Holiday said, but "none of it was left in the picture. And very damn little of me. I know I wore a white dress for a number I did... and that was cut out of the picture." She recorded "The Blues Are Brewin'" for the film's soundtrack. Other songs included in the movie are "Do You Know What It Means to Miss New Orleans?" and "Farewell to Storyville". Holiday's drug addictions were a problem on the set. She earned more than one thousand dollars per week from club ventures but spent most of it on heroin. Her lover, Joe Guy, traveled to Hollywood while Holiday was filming and supplied her with drugs. Guy was banned from the set when he was found there by Holiday's manager, Joe Glaser.

By the late 1940s, Holiday had begun recording a number of slow, sentimental ballads. Metronome expressed its concerns in 1946 about "Good Morning Heartache", saying, "there's a danger that Billie's present formula will wear thin, but up to now it's wearing well." The New York Herald Tribune reported of a concert in 1946 that her performance had little variation in melody and no change in tempo.

1947–1952: Legal issues and Carnegie Hall concert
By 1947, Holiday was at her commercial peak, having made $250,000 in the three previous years. She was ranked second in the DownBeat poll for 1946 and 1947, her highest ranking in that poll. She was ranked fifth in Billboard's annual college poll of "girl singers" on July 6, 1947 (Jo Stafford was first). In 1946, Holiday won the Metronome magazine popularity poll.

On May 16, 1947, Holiday was arrested for possession of narcotics in her New York apartment. On May 27, she was in court. "It was called 'The United States of America versus Billie Holiday'. And that's just the way it felt", she recalled. During the trial, she heard that her lawyer would not come to the trial to represent her. "In plain English, that meant no one in the world was interested in looking out for me," she said. Dehydrated and unable to hold down food, she pleaded guilty and asked to be sent to the hospital. The district attorney spoke in her defense, saying, "If your honor please, this is a case of a drug addict, but more serious, however, than most of our cases, Miss Holiday is a professional entertainer and among the higher rank as far as income was concerned." She was sentenced to Alderson Federal Prison Camp in West Virginia. The drug possession conviction caused her to lose her New York City Cabaret Card, preventing her working anywhere that sold alcohol; thereafter, she performed in concert venues and theaters.

Holiday was released early (on March 16, 1948) because of good behavior. When she arrived at Newark, her pianist Bobby Tucker and her dog Mister were waiting. The dog leaped at Holiday, knocking off her hat, and tackling her to the ground. "He began lapping me and loving me like crazy", she said. A woman thought the dog was attacking Holiday. She screamed, a crowd gathered, and reporters arrived. "I might just as well have wheeled into Penn Station and had a quiet little get-together with the Associated Press, United Press, and International News Service", she said.

Ed Fishman (who fought with Joe Glaser to be Holiday's manager) thought of a comeback concert at Carnegie Hall. Holiday hesitated, unsure audiences would accept her after the arrest. She gave in and agreed to appear. On March 27, 1948, Holiday played Carnegie Hall to a sold-out crowd. Two thousand seven hundred tickets were sold in advance, a record at the time for the venue. Her popularity was unusual because she did not have a current hit record. Her last record to reach the charts was "Lover Man" in 1945. Holiday sang 32 songs at the Carnegie concert by her count, including Cole Porter's "Night and Day" and her 1930s hit, "Strange Fruit". During the show, someone sent her a box of gardenias. "My old trademark", Holiday said. "I took them out of box and fastened them smack to the side of my head without even looking twice." There was a hatpin in the gardenias and Holiday unknowingly stuck it into the side of her head. "I didn't feel anything until the blood started rushing down in my eyes and ears", she said. After the third curtain call, she passed out.

On April 27, 1948, Bob Sylvester and her promoter Al Wilde arranged a Broadway show for her. Titled Holiday on Broadway, it sold out. "The regular music critics and drama critics came and treated us like we were legit", she said. But it closed after three weeks.

Holiday was arrested again on January 22, 1949, in her room at the Hotel Mark Twain in San Francisco by George Hunter White. Holiday said she began using hard drugs in the early 1940s. She married trombonist Jimmy Monroe on August 25, 1941. While still married, she became involved with trumpeter Joe Guy, her drug dealer. She divorced Monroe in 1947 and also split with Guy.

In October 1949, Holiday recorded "Crazy He Calls Me", which was inducted into the Grammy Hall of Fame in 2010. Gabler said the hit was her most successful recording for Decca after "Lover Man". The charts of the 1940s did not list songs outside the top 30, making it impossible to recognize minor hits. By the late 1940s, despite her popularity and concert power, her singles were little played on radio, perhaps because of her reputation.

In 1948, Holiday played at the Ebony Club, which was against the law. Her manager, John Levy, was convinced he could get her card back and allowed her to open without one. "I opened scared", Holiday said, "[I was] expecting the cops to come in any chorus and carry me off. But nothing happened. I was a huge success."

Holiday recorded Gershwin's "I Loves You, Porgy" in 1948. In 1950, Holiday appeared in the Universal short film Sugar Chile Robinson, Billie Holiday, Count Basie and His Sextet, singing "God Bless the Child" and "Now, Baby or Never".

The loss of her cabaret card reduced Holiday's earnings. She had not received proper record royalties until she joined Decca, so her main revenue was club concerts. The problem worsened when Holiday's records went out of print in the 1950s. She seldom received royalties in her later years. In 1958, she received a royalty of only $11. Her lawyer in the late 1950s, Earle Warren Zaidins, registered with BMI only two songs she had written or co-written, costing her revenue.

1952–1959: Lady Sings the Blues
By the 1950s, Holiday's drug use, drinking, and relationships with abusive men caused her health to deteriorate. She appeared on the ABC reality series The Comeback Story to discuss attempts to overcome her poor choices.

Holiday first toured Europe in 1954 as part of a Leonard Feather package. The Swedish impresario Nils Hellstrom initiated the "Jazz Club U.S.A." (after the Leonard Feather radio show) tour starting in Stockholm in January 1954 and then Germany, Netherlands, Paris and Switzerland. The tour party was Holiday, Buddy DeFranco, Red Norvo, Carl Drinkard, Elaine Leighton (de) (nl) (1926–2012), Sonny Clark, Beryl Booker, Jimmy Raney and Red Mitchell. A recording of a live set in Germany was released as Lady Love – Billie Holiday.

Holiday's autobiography, Lady Sings the Blues, was ghostwritten by William Dufty and published in 1956. Dufty, a New York Post writer and editor then married to Holiday's close friend Maely Dufty, wrote the book quickly from a series of conversations with the singer in the Duftys' 93rd Street apartment. He also drew on the work of earlier interviewers and intended to let Holiday tell her story in her own way. In his 2015 study, Billie Holiday: The Musician and the Myth, John Szwed argued that Lady Sings the Blues is a generally accurate account of her life, but that co-writer Dufty was forced to water down or suppress material by the threat of legal action. According to the reviewer Richard Brody, "Szwed traces the stories of two important relationships that are missing from the book—with Charles Laughton, in the 1930s, and with Tallulah Bankhead, in the late 1940s—and of one relationship that's sharply diminished in the book, her affair with Orson Welles around the time of Citizen Kane. The film version of the book was released in 1972, with Diana Ross playing the role of Holiday.

To accompany her autobiography, Holiday released the LP Lady Sings the Blues in June 1956. The album featured four new tracks, "Lady Sings the Blues", "Too Marvelous for Words", "Willow Weep for Me", and "I Thought About You", and eight new recordings of her biggest hits to date. The re-recordings included "Trav'lin' Light" "Strange Fruit" and "God Bless the Child". A review of the album was published by Billboard magazine on December 22, 1956, calling it a worthy musical complement to her autobiography. "Holiday is in good voice now", wrote the reviewer, "and these new readings will be much appreciated by her following". "Strange Fruit" and "God Bless the Child" were called classics, and "Good Morning Heartache", another reissued track on the LP, was also noted favorably.

On November 10, 1956, Holiday performed two concerts before packed audiences at Carnegie Hall. Live recordings of the second Carnegie Hall concert were released on a Verve/HMV album in the UK in late 1961 called The Essential Billie Holiday. The 13 tracks included on this album featured her own songs "I Love My Man", "Don't Explain" and "Fine and Mellow", together with other songs closely associated with her, including "Body and Soul", "My Man", and "Lady Sings the Blues" (her lyrics accompanied a tune by pianist Herbie Nichols). The liner notes for this album were written partly by Gilbert Millstein of the New York Times, who, according to these notes, served as narrator of the Carnegie Hall concerts. Interspersed among Holiday's songs, Millstein read aloud four lengthy passages from her autobiography, Lady Sings the Blues. He later wrote:

"The narration began with the ironic account of her birth in Baltimore – 'Mom and Pop were just a couple of kids when they got married. He was eighteen, she was sixteen, and I was three' – and ended, very nearly shyly, with her hope for love and a long life with 'my man' at her side. It was evident, even then, that Miss Holiday was ill. I had known her casually over the years and I was shocked at her physical weakness. Her rehearsal had been desultory; her voice sounded tinny and trailed off; her body sagged tiredly. But I will not forget the metamorphosis that night. The lights went down, the musicians began to play and the narration began. Miss Holiday stepped from between the curtains, into the white spotlight awaiting her, wearing a white evening gown and white gardenias in her black hair. She was erect and beautiful; poised and smiling. And when the first section of narration was ended, she sang – with strength undiminished – with all of the art that was hers. I was very much moved. In the darkness, my face burned and my eyes. I recall only one thing. I smiled.'"

The critic Nat Hentoff of DownBeat magazine, who attended the Carnegie Hall concert, wrote the remainder of the sleeve notes on the 1961 album. He wrote of Holiday's performance:

"Throughout the night, Billie was in superior form to what had sometimes been the case in the last years of her life. Not only was there assurance of phrasing and intonation; but there was also an outgoing warmth, a palpable eagerness to reach and touch the audience. And there was mocking wit. A smile was often lightly evident on her lips and her eyes as if, for once, she could accept the fact that there were people who did dig her. The beat flowed in her uniquely sinuous, supple way of moving the story along; the words became her own experiences; and coursing through it all was Lady's sound – a texture simultaneously steel-edged and yet soft inside; a voice that was almost unbearably wise in disillusion and yet still childlike, again at the centre. The audience was hers from before she sang, greeting her and saying good-bye with heavy, loving applause. And at one time, the musicians too applauded. It was a night when Billie was on top, undeniably the best and most honest jazz singer alive."

Her performance of "Fine and Mellow" on CBS's The Sound of Jazz program is memorable for her interplay with her long-time friend Lester Young. Both were less than two years from death. Young died in March 1959. Holiday wanted to sing at his funeral, but her request was denied. Also in 1957, she sang as a headliner with Dinah Washington and others in Jazz Under the Stars, a summer concert series that took place at the Wollman Memorial Theater in New York City's Central Park.

When Holiday returned to Europe almost five years later, in 1959, she made one of her last television appearances for Granada television's British Cabaret show, Chelsea at Nine, in London. The show taped what is believed to be the only existing filmed version of Holiday singing "Strange Fruit". Her final studio recordings were made for MGM Records in 1959, with lush backing from Ray Ellis and his Orchestra, who had also accompanied her on the Columbia album Lady in Satin the previous year (see below). The MGM sessions were released posthumously on a self-titled album, later retitled and re-released as Last Recording.

On March 28, 1957, Holiday married Louis McKay, a mob enforcer. McKay, like most of the men in her life, was abusive. They were separated at the time of her death, but McKay had plans to start a chain of Billie Holiday vocal studios, on the model of the Arthur Murray dance schools. Holiday was childless, but she had two godchildren: singer Billie Lorraine Feather (the daughter of Leonard Feather) and Bevan Dufty (the son of William Dufty).

Illness and death
By early 1959, Holiday was diagnosed with cirrhosis of the liver. Although she had initially stopped drinking on her doctor's orders, it was not long before she relapsed. By May 1959, she had lost 20 lb. Her manager, Joe Glaser, jazz critic Leonard Feather, photojournalist Allan Morrison, and the singer's own friends all tried in vain to persuade her to go to a hospital. On May 31, 1959, Holiday was finally taken to Metropolitan Hospital in New York for treatment of both liver and heart disease. While in hospital, narcotics police came to her hospital room and placed her under house arrest for narcotics possession. On July 15, she received last rites. Holiday died at age 44 at 3:10 am on July 17, 1959, of pulmonary edema and heart failure caused by cirrhosis of the liver.

In her final years, Holiday had been progressively swindled out of her earnings by McKay and she died with US$0.70 in the bank ($7.40 in 2023). The story of her burial plot and how it was managed by her estranged husband, Louis McKay, was documented on NPR in 2012. Her funeral was held on July 21, 1959, at the Church of St. Paul the Apostle in Manhattan. She was buried at Saint Raymond's Cemetery in the Bronx. Michael P. Grace ll, a songwriter and theater producer based in Manhattan, paid for the funeral.

Gilbert Millstein of The New York Times, who was the announcer at Holiday's 1956 Carnegie Hall concerts and wrote parts of the sleeve notes for the album The Essential Billie Holiday, described her death in these sleeve notes, dated 1961:

"Billie Holiday died in Metropolitan Hospital, New York, on Friday, July 17, 1959, in the bed in which she had been arrested for illegal possession of narcotics a little more than a month before, as she lay mortally ill; in the room from which a police guard had been removed – by court order – only a few hours before her death. She had been strikingly beautiful, but her talent was wasted. The worms of every kind of excess – drugs were only one – had eaten her. The likelihood exists that among the last thoughts of this cynical, sentimental, profane, generous and greatly talented woman of 44 was the belief that she was to be arraigned the following morning. She would have been, eventually, although possibly not that quickly. In any case, she removed herself finally from the jurisdiction of any court here below."When Holiday died, The New York Times published a short obituary on page 15 without a byline. She left an estate of $1,000 ($10,577 in 2023), and her best recordings from the 1930s were mostly out of print.

Holiday's public stature grew in the following years. In 1961, she was voted to the Down Beat Hall Of Fame, and soon after Columbia reissued nearly one hundred of her early records. In 1972, Diana Ross's portrayal of Holiday in Lady Sings the Blues was nominated for an Oscar and won a Golden Globe. Holiday was posthumously nominated for 23 Grammy awards.

Singer Adelaide Hall made a secret visit to Holiday's bedside at the Metropolitan Hospital, on or around June 12, 1959. Hall's spoken account of her visit was captured on tape by the journalist Max Jones in 1988. Hall's long-time friend, Iain Cameron Williams, and author of Hall's biography, also had direct knowledge of the visit. However, he refrained from releasing the information as he only had Hall's one-to-one spoken account and no further backup. In July 2022, after finding transcripts of Max Jones's tape, Williams wrote an article for The Syncopated Times about Hall's secret visit. George Jacobs (in his account of being Frank Sinatra's valet "Mr S", page 151)) claims Sinatra also visited Holiday on her death bed, and promised to supply her with the heroin she desperately wanted.

Legacy
Billie Holiday received several Esquire Magazine awards during her lifetime. Her posthumous awards also include being inducted into the Grammy Hall of Fame, Ertegun Jazz Hall of Fame, Rock and Roll Hall of Fame, and the ASCAP Jazz Wall of Fame. In 1985, a statue of Billie Holiday was erected in Baltimore; the statue was completed in 1993 with additional panels of images inspired by her seminal song "Strange Fruit". The Billie Holiday Monument is located at Pennsylvania and West Lafayette avenues in Baltimore's Upton neighborhood. In 2019, Chirlane McCray announced that New York City would build a statue honoring Holiday near Queens Borough Hall.

Frank O'Hara's poem from 1959, "The Day Lady Died"', concludes with an impression of Holiday performing at the Five Spot Café at the end of her career, and the impact of that performance on her listeners. The song Angel of Harlem by Irish rock band U2, released as a single in December 1988, was written as a homage to Billie Holiday.

Vocal style
Holiday's delivery made her performances recognizable throughout her career. Her improvisation compensated for lack of musical education. Holiday said that she always wanted her voice to sound like an instrument and some of her influences were Louis Armstrong and the singer Bessie Smith. Her last major recording, a 1958 album entitled Lady in Satin, features the backing of a 40-piece orchestra conducted and arranged by Ray Ellis, who said of the album in 1997:

"I would say that the most emotional moment was her listening to the playback of 'I'm a Fool to Want You'. There were tears in her eyes ... After we finished the album I went into the control room and listened to all the takes. I must admit I was unhappy with her performance, but I was just listening musically instead of emotionally. It wasn't until I heard the final mix a few weeks later that I realized how great her performance really was."

Frank Sinatra was influenced as a young man by her performances on 52nd Street. He told Ebony magazine in 1958 about her impact:

"With few exceptions, every major pop singer in the US during her generation has been touched in some way by her genius. It is Billie Holiday who was, and still remains, the greatest single musical influence on me. Lady Day is unquestionably the most important influence on American popular singing in the last twenty years."

Films and plays about Holiday
The biographical film Lady Sings the Blues, loosely based on Holiday's autobiography, was released in 1972 and was nominated for five Academy Awards, including Diana Ross for Best Actress. Another film, The United States vs. Billie Holiday, starred Andra Day and was released in 2021. It is based on the book Chasing the Scream by Johann Hari. Director Lee Daniels saw how Holiday was portrayed in the 1972 biopic, and wanted to show her legacy as "a civil rights leader [ ... ] not just a drug addict or a jazz singer". The film also depicts Holiday's bisexuality and relationship with Tallulah Bankhead. Day was nominated for the Academy Award for Best Actress for her performance and won a Golden Globe for Best Performance by an Actress in a Motion Picture - Drama in 2021.

Holiday is the primary character in the play Lady Day at Emerson's Bar and Grill, with music by Lanie Robertson. It takes place in South Philadelphia in March 1959. It premiered in 1986 at the Alliance Theatre and has been revived several times. A Broadway production starring Audra McDonald was filmed and broadcast on HBO in 2016; McDonald received an Emmy Award nomination. In 2014, she received a Tony Award win. Billie is a 2019 documentary film based on interviews in the 1970s by Linda Lipnack Kuehl, who was researching a book on Holiday that was never completed.

Billie Holiday was also portrayed by actress Paula Jai Parker in Touched by an Angel 2000 episode "God Bless the Child".

Discography
Billie Holiday recorded extensively for four labels: Columbia Records, which issued her recordings on its subsidiary labels Brunswick Records, Vocalion Records and OKeh Records, from 1933 through 1942; Commodore Records in 1939 and 1944; Decca Records from 1944 through 1950; briefly for Aladdin Records in 1951; Verve Records and on its earlier imprint Clef Records from 1952 through 1957, then again for Columbia Records from 1957 to 1958 and finally for MGM Records in 1959. Many of Holiday's recordings appeared on 78-rpm records prior to the long-playing vinyl record era, and only Clef, Verve, and Columbia issued albums during her lifetime that were not compilations of previously released material. Many compilations have been issued since her death, as well as comprehensive box sets and live recordings.

Hit records
In 1986, Joel Whitburn's company Record Research compiled information on the popularity of recordings released from the era predating rock and roll and created pop charts dating back to the beginning of the commercial recording industry. The company's findings were published in the book Pop Memories 1890–1954. Several of Holiday's records are listed on the pop charts Whitburn created.

Holiday began her recording career on a high note with her first major release, "Riffin' the Scotch", of which 5,000 copies were sold. It was released under the name "Benny Goodman & His Orchestra" in 1933.

Most of Holiday's early successes were released under the name "Teddy Wilson & His Orchestra". During her stay in Wilson's band, Holiday would sing a few bars and then other musicians would have a solo. Wilson, one of the most influential jazz pianists of the swing era, accompanied Holiday more than any other musician. He and Holiday issued 95 recordings together.

In July 1936, Holiday began releasing sides under her own name. These songs were released under the band name "Billie Holiday & Her Orchestra". Most noteworthy, the popular jazz standard "Summertime" sold well and was listed on the pop charts of the time at number 12, the first time the jazz standard charted. Only Billy Stewart's R&B version of "Summertime" reached a higher chart placement than Holiday's, charting at number 10 thirty years later in 1966.

Holiday had 16 best-selling songs in 1937, making the year her most commercially successful. It was in this year that Holiday scored her sole number one hit as a featured vocalist on the available pop charts of the 1930s, "Carelessly". The hit "I've Got My Love to Keep Me Warm", was also recorded by Ray Noble, Glen Gray and Fred Astaire, whose rendering was a bestseller for weeks. Holiday's version ranked 6 on the year-end single chart available for 1937.

In 1939, Holiday recorded her biggest selling record, "Strange Fruit" for Commodore, charting at number 16 on the available pop chart for the 1930s.

In 1940, Billboard began publishing its modern pop charts, which included the Best Selling Retail Records chart, the precursor to the Hot 100. None of Holiday's songs placed on the modern pop charts, partly because Billboard only published the first ten slots of the charts in some issues. Minor hits and independent releases had no way of being spotlighted.

"God Bless the Child", which went on to sell over a million copies, ranked number 3 on Billboard's year-end top songs of 1941.

On October 24, 1942, Billboard began issuing its R&B charts. Two of Holiday's songs placed on the chart, "Trav'lin' Light" with Paul Whiteman, which topped the chart, and "Lover Man", which reached number 5. "Trav'lin' Light" also reached 18 on Billboard's year-end chart.

Studio LPs

 * Billie Holiday Sings (1952)
 * An Evening with Billie Holiday (1953)
 * Billie Holiday (1954)
 * Music for Torching (1955)
 * Velvet Mood (1956)
 * Lady Sings the Blues (1956)
 * Body and Soul (1957)
 * Songs for Distingué Lovers (1957)
 * Stay with Me (1958)
 * All or Nothing at All (1958)
 * Lady in Satin (1958)
 * Last Recording (1959)

Theatrical films

 * 1933: The Emperor Jones, appeared as an extra
 * 1935: Symphony in Black, short (with Duke Ellington)
 * 1947: New Orleans
 * 1950: 'Sugar Chile' Robinson, Billie Holiday, Count Basie and His Sextet

Television appearances
(1) = Available on audio (2) = Available on DVD

General references
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 * Note: Keith, the author, was, at the time, Editor of the Pittsburgh Courier.

title: Billie Holiday