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. 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.

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.

The derivative at different points of a differentiable function. In this case, the derivative is equal to

\sin \left(x^{2}\right)+2x^{2}\cos \left(x^{2}\right)

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. Higher order derivatives are used in physics; for example, the first derivative with respect to time of the position of a moving object is its velocity, and the second derivative is its acceleration.

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 ⁠ $a$ ⁠, and the limit $L=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}$ exists. This means that, for every positive real number ⁠ $\varepsilon$ ⁠, 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.

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. 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 $f$ .

For example, let $f$ be the squaring function: $f(x)=x^{2}$ . Then the quotient in the definition of the derivative is ${\frac {f(a+h)-f(a)}{h}}={\frac {(a+h)^{2}-a^{2}}{h}}={\frac {a^{2}+2ah+h^{2}-a^{2}}{h}}=2a+h.$ 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: ⁠ $f'(x)=2x$ ⁠.

The ratio in the definition of the derivative is the slope of the line through two points on the graph of the function ⁠ $f$ ⁠, 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 ${\textstyle {\frac {df}{dx}}(a)}$ 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 $f'(x)=\operatorname {st} \left({\frac {f(x+dx)-f(x)}{dx}}\right)$ for an arbitrary infinitesimal ⁠ $dx$ ⁠, where $\operatorname {st}$ 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, ${\begin{aligned}f'(x)&=\operatorname {st} \left({\frac {x^{2}+2x\cdot dx+(dx)^{2}-x^{2}}{dx}}\right)\\&=\operatorname {st} \left({\frac {2x\cdot dx+(dx)^{2}}{dx}}\right)\\&=\operatorname {st} \left({\frac {2x\cdot dx}{dx}}+{\frac {(dx)^{2}}{dx}}\right)\\&=\operatorname {st} \left(2x+dx\right)\\&=2x.\end{aligned}}$

Continuity and differentiability

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

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.

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