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width=150 length=150 The graph of a function, drawn in black, and a tangent line to that function, drawn in red.  The slope of the tangent line is equal to the derivative of the function at the marked point.

In [calculus] (a branch of [mathematics] ) the derivative is a measure of how a [function] changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity at which the vehicle is traveling. Conversely, the [integral] of the velocity over time is how much the vehicle's position changes from the time when the integral begins to the time when the integral ends.

The derivative of a function at a chosen input value describes the best [linear approximation] of the function near that input value. For a [real-valued function] of a single real variable, the derivative at a point equals the [slope] of the [tangent line] to the [graph of the function] at that point. In higher dimensions, the derivative of a function at a point is a [linear transformation] called the [linearization] . A closely related notion is the [differential of a function] .

The process of finding a derivative is called differentiation . The reverse process is called [antidifferentiation] . The [fundamental theorem of calculus] states that antidifferentiation is the same as [integration] .

Differentiation and the derivative

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