Derivative and Differentiability¶

Definition¶

Let $X\subseteq \mathbb{R}\text{,}$ let $x_0$ be an interior point of $X\text{,}$ that is, $(x_0-r,x_0+r)\subseteq X$ for some $r>0\text{,}$ and let $f \colon X \to \mathbb{R}$ be a function. We say that $f$ is differentiable at $x=x_0$ with derivative $L\text{,}$ and write $f'(x_0) = L\text{,}$ if the two-sided limit

\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}

exists and equals the real number $L\text{.}$ If the limit does not exist, we leave $f'(x_0)$ undefined and say that $f$ is not differentiable at $x=x_0\text{.}$ At a boundary point the corresponding one-sided derivative is treated separately.

The derivative can be written using either a moving point $x$ or an increment $h\text{:}$

f'(x_0)=\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0} =\lim_{h\to 0}\frac{f(x_0+h)-f(x_0)}{h},

and the substitution $h=x-x_0$ shows the two forms are equivalent.

Finally, if every point of $X$ is an interior point of $X$ (for example, $X$ an open interval), $f$ is differentiable on $X$ if it is differentiable at every $x_0\in X\text{;}$ at the endpoints of a closed interval one uses the one-sided derivative instead.

Interpretation¶

The derivative measures the instantaneous rate of change of a function at a point. The difference quotient $\frac{f(x)-f(x_0)}{x-x_0}$ is the average rate of change between $x_0$ and $x$ , the slope of a secant line, and the derivative is its limit as $x\to x_0$ $x_{0}$ . Differentiability means that this limit exists as a finite number.

A curve with its tangent line at a point x0; near x0 the curve stays close to the tangent, whose slope is the derivative.

Near the marked point $(x_0,f(x_0))$ the curve and the tangent line are hard to tell apart, and the slope of the tangent is $f'(x_0)$ . Here the curve is falling at $x_0$ , so $f'(x_0)<0$ $0$ .

Let $h$ shrink towards $0$ from either side: the secant turns into the tangent and the difference quotient approaches $f'(x_0)$ . For $|x|$ at $x_0=0$ the two sides disagree.

Differentiability at $x_0$ can be read in two equivalent ways. Geometrically, the graph has a non-vertical tangent line at $(x_0,f(x_0))$ , with slope $f'(x_0)$ . Analytically, in a small neighbourhood of $x_0$ the function is approximated by the tangent line: $f(x)=f(x_0)+f'(x_0)(x-x_0)+r(x)$ with an error $r(x)$ that is small even relative to $x-x_0$ $x_{0}$ , that is, $r(x)/(x-x_0)\to0$ $0$ . The second reading is the one behind the differential.

Differentiability is stronger than continuity: letting $x\to x_0$ in $f(x)-f(x_0)=\frac{f(x)-f(x_0)}{x-x_0}\,(x-x_0)$ gives $f(x)\to f(x_0)$ $f (x_{0})$ . The converse fails at a corner, where the one-sided derivatives differ, and at a vertical tangent: for $\sqrt[3]{x}$ at $0$ the difference quotient is $1/\sqrt[3]{x^2}$ , which tends to $+\infty$ .

The approximation reading returns in Math III as the total derivative of a function of several variables.

Key Results¶

Differentiability implies continuity.
The derivative is unique when it exists.
Non-differentiability often signals sharp changes in behaviour.

Examples¶

The function $f(x)=x^2$ is differentiable everywhere.
The function $f(x)=|x|$ is not differentiable at $x=0\text{.}$

Prerequisites¶

Limit at a Point

Sources¶

Math I: Calculus · Lecture 6

Keywords: derivative, differentiable function, difference quotient Symbols: element of, infinity, real numbers, square root, subset

Last updated: 22 October 2026 19:04