Derivative

The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input. 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.^[1] The process of finding a derivative is called differentiation.

For other uses, see Derivative (disambiguation).

There are multiple different notations for differentiation, two of the most commonly used being Leibniz notation and prime notation. Leibniz notation, named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas prime notation is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leibniz notation by adding superscripts to the differentials, and in prime notation by adding additional prime marks. The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances.

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.

Definition

As a limit

A function of a real variable $f(x)$ is differentiable at a point $a$ of its domain, if its domain contains an open interval containing $a$ , and the limit $L=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}$ exists.^[2] This means that, for every positive real number $\varepsilon$ , 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. This is an example of the (ε, δ)-definition of limit.^[3]

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 $a$ . Multiple notations for the derivative exist.^[4] The derivative of $f$ at $a$ can be denoted $f'(a)$ , read as " $f$ prime of $a$ "; or it can be denoted ${\textstyle {\frac {df}{dx}}(a)}$ , read as "the derivative of $f$ with respect to $x$ at $a$ " or " $df$ by (or over) $dx$ at $a$ ". See § Notation below. If $f$ is a function that has a derivative at every point in its domain, then a function can be defined by mapping every point $x$ to the value of the derivative of $f$ at $x$ . This function is written $f'$ and is called the derivative function or the derivative of $f$ . The function $f$ sometimes has a derivative at most, but not all, points of its domain. The function whose value at $a$ equals $f'(a)$ whenever $f'(a)$ is defined and elsewhere is undefined is also called the derivative of $f$ . It is still a function, but its domain may be smaller than the domain of $f$ .^[5]

For example, let $f$ be the squaring function: $f(x)=x^{2}$ . Then the quotient in the definition of the derivative is^[6] ${\frac {f(a+h)-f(a)}{h}}={\frac {(a+h)^{2}-a^{2}}{h}}={\frac {a^{2}+2ah+h^{2}-a^{2}}{h}}=2a+h.$ The division in the last step is valid as long as $h\neq 0$ . The closer $h$ is to $0$ , the closer this expression becomes to the value $2a$ . The limit exists, and for every input $a$ the limit is $2a$ . So, the derivative of the squaring function is the doubling function: $f'(x)=2x$ .

:
${\frac {d}{dx}}x^{a}=ax^{a-1}$

Derivatives of powers

Functions of , natural logarithm, and logarithm with general base:
${\frac {d}{dx}}e^{x}=e^{x}$

${\frac {d}{dx}}a^{x}=a^{x}\ln(a)$ , for $a>0$

${\frac {d}{dx}}\ln(x)={\frac {1}{x}}$ , for $x>0$

${\frac {d}{dx}}\log _{a}(x)={\frac {1}{x\ln(a)}}$ , for $x,a>0$

exponential

:
${\frac {d}{dx}}\sin(x)=\cos(x)$

${\frac {d}{dx}}\cos(x)=-\sin(x)$

${\frac {d}{dx}}\tan(x)=\sec ^{2}(x)={\frac {1}{\cos ^{2}(x)}}=1+\tan ^{2}(x)$

Trigonometric functions

:
${\frac {d}{dx}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}}$ , for $-1<x<1$

${\frac {d}{dx}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}}$ , for $-1<x<1$

${\frac {d}{dx}}\arctan(x)={\frac {1}{1+x^{2}}}$

Inverse trigonometric functions

Higher-order derivatives

Higher order derivatives means that a function is differentiated repeatedly. Given that $f$ is a differentiable function, the derivative of $f$ is the first derivative, denoted as $f'$ . The derivative of $f'$ is the second derivative, denoted as $f''$ , and the derivative of $f''$ is the third derivative, denoted as $f'''$ . By continuing this process, if it exists, the $n$ -th derivative as the derivative of the $(n-1)$ -th derivative or the derivative of order $n$ . As has been discussed above, the generalization of derivative of a function $f$ may be denoted as $f^{(n)}$ .^[31] A function that has $k$ successive derivatives is called $k$ times differentiable. If the $k$ -th derivative is continuous, then the function is said to be of differentiability class $C^{k}$ .^[32] A function that has infinitely many derivatives is called infinitely differentiable or smooth.^[33] One example of the infinitely differentiable function is polynomial; differentiate this function repeatedly results the constant function, and the infinitely subsequent derivative of that function are all zero.^[34]

In one of its applications, the higher-order derivatives may have specific interpretations in physics. Suppose that a function represents the position of an object at the time. The first derivative of that function is the velocity of an object with respect to time, the second derivative of the function is the acceleration of an object with respect to time,^[28] and the third derivative is the jerk.^[35]

An important generalization of the derivative concerns of complex variables, such as functions from (a domain in) the complex numbers $\mathbb {C}$ to $\mathbb {C}$ . The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition.^[47] If $\mathbb {C}$ is identified with $\mathbb {R} ^{2}$ by writing a complex number $z$ as $x+iy$ , then a differentiable function from $\mathbb {C}$ to $\mathbb {C}$ is certainly differentiable as a function from $\mathbb {R} ^{2}$ to $\mathbb {R} ^{2}$ (in the sense that its partial derivatives all exist), but the converse is not true in general: the complex derivative only exists if the real derivative is complex linear and this imposes relations between the partial derivatives called the Cauchy–Riemann equations – see holomorphic functions.^[48]

complex functions

Another generalization concerns functions between . Intuitively speaking such a manifold $M$ is a space that can be approximated near each point $x$ by a vector space called its tangent space: the prototypical example is a smooth surface in $\mathbb {R} ^{3}$ . The derivative (or differential) of a (differentiable) map $f:M\to N$ between manifolds, at a point $x$ in $M$ , is then a linear map from the tangent space of $M$ at $x$ to the tangent space of $N$ at $f(x)$ . The derivative function becomes a map between the tangent bundles of $M$ and $N$ . This definition is used in differential geometry.^[49]

differentiable or smooth manifolds

Differentiation can also be defined for maps between , such as Banach space, in which those generalizations are the Gateaux derivative and the Fréchet derivative.^[50]

vector space

One deficiency of the classical derivative is that very many functions are not differentiable. Nevertheless, there is a way of extending the notion of the derivative so that all functions and many other functions can be differentiated using a concept known as the weak derivative. The idea is to embed the continuous functions in a larger space called the space of distributions and only require that a function is differentiable "on average".^[51]

continuous

Properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology; an example is . Here, it consists of the derivation of some topics in abstract algebra, such as rings, ideals, field, and so on.^[52]

differential algebra

The discrete equivalent of differentiation is . The study of differential calculus is unified with the calculus of finite differences in time scale calculus.^[53]

finite differences

The involves the function that is defined for the integers by the prime factorization. This is an analogy with the product rule.^[54]

arithmetic derivative

The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as a linear approximation of the function at that point.

Integral

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Derivative"

: "Newton, Leibniz, and Usain Bolt"

Khan Academy

"Derivative". MathWorld.

Weisstein, Eric W.

from Wolfram Alpha.