The derivative is the central concept of differential calculus — the precise formulation of the idea that a function has an instantaneous rate of change. Where a [limit](/page/Limit) captures the behaviour of a quantity as a parameter tends to some value, the derivative applies this mechanism to the difference quotient of a function, extracting the slope of the tangent line from the slopes of secant lines. Every major application of calculus — optimisation, differential equations, Taylor approximation, the [Fundamental Theorem of Calculus](/theorems/632) linking differentiation to [integration](/page/Integral) — depends on the derivative. This page develops the derivative of a real-valued function of one real variable: its definition, its algebraic properties, the mean value theorems that give it global reach, and the higher-order theory culminating in Taylor's theorem.

[motivation] ### Rates of Change and the Tangent Problem Many quantities in mathematics and the sciences vary with respect to one another: position changes with time, pressure changes with volume, area changes with the length of a side. The fundamental question of differential calculus is: *at what rate does one quantity change with respect to another, at a specific instant?* The difficulty is that "instantaneous rate of change" is not a directly observable quantity. Over a time interval $[t_0, t_0 + h]$, one can measure the average velocity of a particle as the ratio $\Delta x / \Delta t = (x(t_0 + h) - x(t_0))/h$. But this is a rate over an interval, not at a point. Taking $h$ smaller gives a better approximation, but setting $h = 0$ produces $0/0$ — the ratio is undefined. The derivative resolves this by defining the instantaneous rate as the [limit](/page/Limit) of the average rate as $h \to 0$, provided the limit exists. ### What Fails Without the Derivative Without the derivative, one cannot formulate the condition for a function to have a local extremum, determine when two functions grow at the same rate, approximate a function by polynomials, or relate the rate of change of a quantity to its accumulation. The mean value theorem — the bridge between local information (the derivative at each point) and global information (the change over an interval) — requires differentiability as a hypothesis. Taylor's theorem, which approximates smooth [functions](/page/Function) by polynomials with explicit error bounds, is built entirely on iterated differentiation. And the [Fundamental Theorem of Calculus](/theorems/632), which connects the [integral](/page/Integral) to antidifferentiation, makes differentiation indispensable for computing integrals. ### The Geometric Picture Geometrically, the derivative of $f$ at $a$ is the slope of the unique line that best approximates the graph of $f$ near $(a, f(a))$. The secant line through $(a, f(a))$ and $(a+h, f(a+h))$ has slope $(f(a+h) - f(a))/h$; as $h \to 0$, these secant lines rotate toward a limiting position — the tangent line — and the derivative is the slope of this tangent. The existence of the derivative is thus equivalent to the existence of a well-defined tangent direction, which fails at corners, cusps, and points of vertical tangency. [/motivation]

## Definition

The derivative is defined as a limit of difference quotients. The domain of the function must be such that the limit can be formed — the point must be a limit point of the domain from both sides (for the full derivative) or from one side (for one-sided derivatives).

[definition: Derivative] Let $E \subseteq \mathbb{R}$, let $a \in E$ be a limit point of $E$, and let $f : E \to \mathbb{R}$. The **derivative of $f$ at $a$** is \begin{align*} f'(a) &:= \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}, \end{align*} provided this [limit](/page/Limit) exists and is finite. Equivalently, \begin{align*} f'(a) &= \lim_{x \to a} \frac{f(x) - f(a)}{x - a}. \end{align*} If $f'(a)$ exists, $f$ is said to be **differentiable at $a$**. If $f$ is differentiable at every point of an [open set](/page/Open%20Set) $U \subseteq E$, $f$ is **differentiable on $U$**, and the function \begin{align*} f' : U &\to \mathbb{R} \\ a &\mapsto f'(a) \end{align*} is called the **derivative of $f$**. [/definition]

The two formulations are related by the substitution $x = a + h$. The limit $h \to 0$ is taken over all $h \neq 0$ such that $a + h \in E$; the limit $x \to a$ is taken over all $x \in E \setminus \{a\}$. In either case, the definition asks for a limit of the slopes of secant lines through the fixed point $(a, f(a))$.

An equivalent characterisation — often more useful for proofs — reformulates differentiability as a linear approximation property. The function $f$ is differentiable at $a$ with derivative $f'(a) = L$ if and only if there exists a function $\varepsilon : E \to \mathbb{R}$ with $\varepsilon(x) \to 0$ as $x \to a$ such that

\begin{align*} f(x) &= f(a) + L(x - a) + \varepsilon(x)(x - a) \quad \text{for all } x \in E. \end{align*}

This says that $f$ is approximated near $a$ by the affine function $x \mapsto f(a) + L(x-a)$ — the tangent line — with an error that is $o(|x-a|)$ as $x \to a$. The derivative $L = f'(a)$ is the unique real number for which such an approximation exists. This linear-approximation viewpoint is the starting point for the generalisation to several variables, where the derivative becomes a [linear map](/page/Linear%20Map) rather than a number.

[example: Derivative Of A Power Function] Let $f : \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = x^n$ for a fixed $n \in \mathbb{N}$. For any $a \in \mathbb{R}$, the difference quotient is \begin{align*} \frac{f(a+h) - f(a)}{h} &= \frac{(a+h)^n - a^n}{h}. \end{align*} Using the factorisation $u^n - v^n = (u - v)(u^{n-1} + u^{n-2}v + \cdots + v^{n-1})$ with $u = a+h$ and $v = a$: \begin{align*} \frac{(a+h)^n - a^n}{h} &= (a+h)^{n-1} + (a+h)^{n-2}a + \cdots + a^{n-1}. \end{align*} This is a sum of $n$ terms. As $h \to 0$, each $(a+h)^{n-1-k} a^k \to a^{n-1}$, so \begin{align*} f'(a) &= \lim_{h \to 0} \sum_{k=0}^{n-1} (a+h)^{n-1-k} a^k = n a^{n-1}. \end{align*} The power rule $f'(a) = na^{n-1}$ extends to all real exponents $n \in \mathbb{R}$ (for $a > 0$) via the exponential function: if $f(x) = x^n = e^{n \ln x}$, the chain rule gives $f'(x) = n x^{n-1}$. [/example]

## Differentiability and [Continuity](/page/Continuity)

Differentiability is a stronger condition than continuity: every differentiable function is continuous, but the converse fails dramatically. Understanding this gap is essential for knowing when the tools of differential calculus apply.

The relationship between the two concepts reflects the geometric distinction between functions whose graphs have well-defined tangent lines and functions whose graphs are too rough or irregular for any linear approximation to work.

[theorem: Differentiability Implies Continuity] Let $E \subseteq \mathbb{R}$, $a \in E$ a limit point, and $f : E \to \mathbb{R}$ differentiable at $a$. Then $f$ is continuous at $a$. [/theorem]

The proof uses the decomposition $f(x) - f(a) = \frac{f(x) - f(a)}{x - a} \cdot (x - a)$. As $x \to a$, the first factor tends to $f'(a)$ (a finite number) and the second factor tends to $0$, so their product tends to $0$, giving $f(x) \to f(a)$.

The converse fails: continuity does not imply differentiability. The simplest example is the absolute value function, which is continuous everywhere but not differentiable at the origin.

[example: Non-Differentiability Of The Absolute Value] The function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = |x|$ is continuous at $0$ (since $|f(x) - f(0)| = |x| \to 0$) but not differentiable at $0$. The left and right difference quotients give different limits: \begin{align*} \lim_{h \to 0^+} \frac{|h| - 0}{h} &= \lim_{h \to 0^+} \frac{h}{h} = 1, \\ \lim_{h \to 0^-} \frac{|h| - 0}{h} &= \lim_{h \to 0^-} \frac{-h}{h} = -1. \end{align*} Since the one-sided limits disagree, the two-sided limit does not exist, and $f'(0)$ is undefined. Geometrically, the graph of $|x|$ has a corner at the origin: the left half has slope $-1$ and the right half has slope $+1$, so there is no single tangent line. [/example]

[example: A Continuous Nowhere-Differentiable Function] The failure of the converse is far worse than isolated corners. Weierstrass (1872) exhibited a continuous function $f : \mathbb{R} \to \mathbb{R}$ that is differentiable at *no* point. One such construction is \begin{align*} f(x) &= \sum_{n=0}^\infty a^n \cos(b^n \pi x), \end{align*} where $0 < a < 1$, $b$ is a positive odd integer, and $ab > 1 + \frac{3\pi}{2}$. The [series](/page/Series) [converges uniformly](/page/Uniform%20Convergence) (by the Weierstrass $M$-test, since $|a^n \cos(b^n \pi x)| \leq a^n$ and $\sum a^n < \infty$), so $f$ is continuous. But the condition $ab > 1$ ensures that the oscillations at scale $b^{-n}$ grow faster than the damping factor $a^n$ contracts them, preventing the difference quotient from converging at any point. The existence of such functions demonstrates that continuity and differentiability are genuinely different properties — one cannot infer differentiability from continuity alone, even at a single point. [/example]

## Algebraic Rules

Computing derivatives from the definition for each new function would be impractical. The algebraic rules — the sum rule, product rule, quotient rule, and chain rule — reduce the differentiation of complicated expressions to the differentiation of their building blocks. These rules, together with the derivatives of the elementary functions, suffice to differentiate any expression built from polynomials, rational functions, exponentials, logarithms, and trigonometric functions.