Derivative

This article is about the term as used in calculus. For a less technical overview of the subject, see differential calculus. For other uses, see Derivative (disambiguation).

File:Tangent to a curve.svg

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

In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.

The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.

Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.

The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.

Definition

A function of a real variable f(x) is differentiable at a point a of its domain, if its domain contains an open interval I containing a, and the limit

$L=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}$

exists. This means that, for every positive real number $\varepsilon$ (even very small), there exists a positive real number $\delta$ such that, for every h such that $|h|<\delta$ and $h\neq 0$ then $f(a+h)$ is defined, and

$\left|L-{\frac {f(a+h)-f(a)}{h}}\right|<\varepsilon ,$

where the vertical bars denote the absolute value (see (ε, δ)-definition of limit).

If the function f is differentiable at a, that is if the limit L exists, then this limit is called the derivative of f at a, and denoted $f'(a)$ (read as "f prime of a") or ${\textstyle {\frac {df}{dx}}(a)}$ (read as "the derivative of f with respect to x at a" or "df by (or over) dx at a"); see § Notation (details), below.

Continuity and differentiability

File:Right-continuous.svg

This function does not have a derivative at the marked point, as the function is not continuous there (specifically, it has a jump discontinuity).

If f is differentiable at a, then f must also be continuous at a. As an example, choose a point a and let f be the step function that returns the value 1 for all x less than a, and returns a different value 10 for all x greater than or equal to a. f cannot have a derivative at a. If h is negative, then $a+h$ is on the low part of the step, so the secant line from a to $a+h$ is very steep; as h tends to zero, the slope tends to infinity. If h is positive, then $a+h$ is on the high part of the step, so the secant line from a to $a+h$ has slope zero. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist.

File:Absolute value.svg

The absolute value function is continuous but fails to be differentiable at x = 0 since the tangent slopes do not approach the same value from the left as they do from the right.

However, even if a function is continuous at a point, it may not be differentiable there. For example, the absolute value function given by $f(x)=|x|$ is continuous at