See derived function

• Czech: derivace
• Finnish: derivaatta
• French: dérivée
• German: Ableitung
• Italian: derivata
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• ttbc Spanish: derivativo (1-4)

In calculus, a branch of mathematics, the derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point. For example, the derivative of the position or distance of a car at some point in time is the instantaneous velocity, or instantaneous speed (respectively), at which that car is traveling (conversely the integral of the velocity is the car's position).

A closely related notion is the differential of a function.

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.

The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration.

Differentiation and the derivative

Differentiation is a method to compute the rate at which a quantity, y, changes with respect to the change in another quantity, x, upon which it is dependent. This rate of change is called the derivative of y with respect to x. In more precise language, the dependency of y on x means that y is a function of x. If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point. This functional relationship is often denoted y = f(x), where f denotes the function.

The simplest case is when y is a linear function of x, meaning that the graph of y against x is a straight line. In this case, y = f(x) = m x + c, for real numbers m and c, and the slope m is given by

m= =

where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in." This formula is true because

y + Δy = f(x+ Δx) = m (x + Δx) + c = m x + c + m Δx = y + mΔx.

It follows that Δy = m Δx.

This gives an exact value for the slope of a straight line. If the function f is not linear (i.e. its graph is not a straight line), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.

The idea, illustrated by Figures 1-3, is to compute the rate of change as the limiting value of the ratio of the differences Δy / Δx as Δx becomes infinitely small.

In Leibniz's notation, such an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written

\frac \,\!

suggesting the ratio of two infinitesimal quantities. (The above expression is pronounced in various ways such as "d y by d x" or "d y over d x". The oral form "d y d x" is often used conversationally, although it may lead to confusion.)

The most common approach to turn this intuitive idea into a precise definition uses limits, but there are other methods, such as non-standard analysis.

Definition via difference quotients

Let y=f(x) be a function of x. In classical geometry, the tangent line at a real number a was the unique line through the point (a, f(a)) which did not meet the graph of f transversally, meaning that the line did not pass straight through the graph. The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at a. The slope of the tangent line is very close to the slope of the line through (a, f(a)) and a nearby point on the graph, for example (a + h, f(a + h)). These lines are called secant lines. A value of h close to zero will give a good approximation to the slope of the tangent line, and smaller values (in absolute value) of h will, in general, give better approximations. The slope of the secant line is the difference between the y values of these points divided by the difference between the x values, that is,

\frac.

This expression is Newton's difference quotient. The derivative is the value of the difference quotient as the secant lines get closer and closer to the tangent line. Formally, the derivative of the function f at a is the limit

f'(a)=\lim_

of the difference quotient as h approaches zero, if this limit exists. If the limit exists, then f is differentiable at a. Here f′ (a) is one of several common notations for the derivative (see below).

Equivalently, the derivative satisfies the property that

\lim_ = 0,

which has the intuitive interpretation (see Figure 1) that the tangent line to f at a gives the best linear approximation

f(a+h) \approx f(a) + f'(a)h

to f near a (i.e., for small h). This interpretation is the easiest to generalize to other settings (see below).

Substituting 0 for h in the difference quotient causes division by zero, so the slope of the tangent line cannot be found directly. Instead, define Q(h) to be the difference quotient as a function of h:

Q(h) = \frac.

Q(h) is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). If f is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from the point h = 0. If the limit \textstyle\lim_ Q(h) exists, meaning that there is a way of choosing a value for Q(0) which makes the graph of Q a continuous function, then the function f is differentiable at the point a, and its derivative at a equals Q(0).

In practice, the existence of a continuous extension of the difference quotient Q(h) to h = 0 is shown by modifying the numerator to cancel h in the denominator. This process can be long and tedious for complicated functions, and many short cuts are commonly used to simplify the process.

Example

The squaring function f(x) = x² is differentiable at x = 3, and its derivative there is 6. This is proven by writing the difference quotient as follows:

= = = = 6 + h.

Then we get the simplified function in the limit:

\lim_ 6 + h = 6 + 0 = 6.

The last expression shows that the difference quotient equals 6 + h when h is not zero and is undefined when h is zero. (Remember that because of the definition of the difference quotient, the difference quotient is always undefined when h is zero.) However, there is a natural way of filling in a value for the difference quotient at zero, namely 6. Hence the slope of the graph of the squaring function at the point (3, 9) is 6, and so its derivative at x = 3 is f '(3) = 6.

More generally, a similar computation shows that the derivative of the squaring function at x = a is f '(a) = 2a.

Continuity and differentiability

• Automatic differentiation
• Differentiability class
• Differintegral
• Integral
• Linearization
• Numerical differentiation
• Techniques for differentiation
• Table of derivatives

External links

• WIMS Function Calculator makes online calculation of derivatives; this software also enables interactive exercises.
• Online Derivatives Calculator.
• Mathematical Assistant on Web online calculation of derivatives, including explanation of steps in the solution.
• Proving Derivatives from First Principles.
• Practice finding derivatives of randomly generated functions

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