derivative

In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to 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. The process of finding a derivative is called differentiation.

There are multiple different notations for differentiation. '' 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 case, 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 , and the limit
$L=\lim_\frach$
exists. This means that, for every positive real number , there exists a positive real number $\delta$ such that, for every $h$ such that $, h, < \delta$ and $h\ne 0$ then $f(a+h)$ is defined, and
$\left, L-\frach\<\varepsilon,$
where the vertical bars denote the absolute value. This is an example of the (ε, δ)-definition of limit.

If the function $f$ is differentiable at , that is if the limit $L$ exists, then this limit is called the ''derivative'' of $f$ at $a$. Multiple notations for the derivative exist. The derivative of $f$ at $a$ can be denoted , read as " prime of "; or it can be denoted , read as "the derivative of $f$ with respect to $x$ at " or " by (or over) $dx$ at ". See ' 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'' . 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 . It is still a function, but its domain may be smaller than the domain of $f$.

For example, let $f$ be the squaring function: $f(x) = x^2$. Then the quotient in the definition of the derivative is
$\frac = \frac = \frac = 2a + h.$
The division in the last step is valid as long as $h \neq 0$. The closer $h$ is to , 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: .

The ratio in the definition of the derivative is the slope of the line through two points on the graph of the function , specifically the points $(a,f(a))$ and $(a+h, f(a+h))$. As $h$ is made smaller, these points grow closer together, and the slope of this line approaches the limiting value, the slope of the tangent to the graph of $f$ at $a$. In other words, the derivative is the slope of the tangent.

Using infinitesimals

One way to think of the derivative $\frac(a)$ is as the ratio of an infinitesimal change in the output of the function $f$ to an infinitesimal change in its input. In order to make this intuition rigorous, a system of rules for manipulating infinitesimal quantities is required. The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals are an extension of the real numbers that contain numbers greater than anything of the form $1 + 1 + \cdots + 1$ for any finite number of terms. Such numbers are infinite, and their reciprocals are infinitesimals. The application of hyperreal numbers to the foundations of calculus is called nonstandard analysis. This provides a way to define the basic concepts of calculus such as the derivative and integral in terms of infinitesimals, thereby giving a precise meaning to the $d$ in the Leibniz notation. Thus, the derivative of $f(x)$ becomes $f'(x) = \operatorname\left( \frac \right)$ for an arbitrary infinitesimal , where $\operatorname$ denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real. Taking the squaring function $f(x) = x^2$ as an example again,
$\begin f'(x) &= \operatorname\left(\frac\right) \\ &= \operatorname\left(\frac\right) \\ &= \operatorname\left(\frac + \frac\right) \\ &= \operatorname\left(2x + dx\right) \\ &= 2x. \end$

Continuity and differentiability

If $f$ is differentiable at , 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 , and returns a different value 10 for all $x$ greater than or equal to $a$. The function $f$ 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. 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 , but it is not differentiable there. If $h$ is positive, then the slope of the secant line from 0 to $h$ is one; if $h$ is negative, then the slope of the secant line from $0$ to $h$ is . This can be seen graphically as a "kink" or a "cusp" in the graph at $x=0$. Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function given by $f(x) = x^$ is not differentiable at $x = 0$. In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative.

Most functions that occur in practice have derivatives at all points or almost every point. Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points. Under mild conditions (for example, if the function is a monotone or a Lipschitz function), this is true. However, in 1872, Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as the Weierstrass function. In 1931, Stefan Banach proved that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions. Informally, this means that hardly any random continuous functions have a derivative at even one point.

Notation

One common way of writing the derivative of a function is Leibniz notation, introduced by Gottfried Wilhelm Leibniz in 1675, which denotes a derivative as the quotient of two differentials, such as $dy$ and . It is still commonly used when the equation $y=f(x)$ is viewed as a functional relationship between dependent and independent variables. The first derivative is denoted by , read as "the derivative of $y$ with respect to ". This derivative can alternately be treated as the application of a differential operator to a function, $\frac = \frac f(x).$ Higher derivatives are expressed using the notation $\frac$ for the $n$-th derivative of $y = f(x)$. These are abbreviations for multiple applications of the derivative operator; for example, $\frac = \frac\Bigl(\frac f(x)\Bigr).$ Unlike some alternatives, Leibniz notation involves explicit specification of the variable for differentiation, in the denominator, which removes ambiguity when working with multiple interrelated quantities. The derivative of a composed function can be expressed using the chain rule: if $u = g(x)$ and $y = f(g(x))$ then $\frac = \frac \cdot \frac.$

Another common notation for differentiation is by using the prime mark in the symbol of a function . This notation, due to Joseph-Louis Lagrange

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