Derivative of a Real Function

Overview

The derivative measures a local rate of change through a limit of secant slopes. A finite derivative is more than a numerical slope: it gives a linear approximation whose relative error vanishes as the input increment shrinks. This interpretation explains why derivatives support both geometric reasoning and reliable small-change calculations.

A derivative is attached to a point and a domain. Differentiating a symbolic expression does not remove the need to check whether the relevant limit exists.

Prerequisites

Know limits, continuity, and the distinction between an equality and an approximation. The notation $r(h)=o(|h|)$ means $r(h)/|h|\to0$ as $h\to0$ through nonzero values. Understanding a line as a linear function of its increment makes the definition more transparent.

Definition and conventions

Suppose $f:I\to\mathbb R$, where $I$ is an open interval containing $a$. Its derivative at $a$ is

provided the limit exists as a finite real number. Equivalently,

One-sided derivatives use positive or negative increments only. An infinite slope is not a derivative in this convention. The differential is the linear map $h\mapsto f'(a)h$, rather than the derivative number itself.

Intuition

Zooming near $(a,f(a))$ makes a differentiable graph increasingly resemble its tangent. The remainder criterion expresses this without relying on an image: after dividing the error by the input size, the error tends to zero. A corner has incompatible slopes from the two sides, so no single linear map describes it.

Examples and non-examples

For $f(x)=x^3$,

hence $f'(a)=3a^2$. At $a=2$, the linear approximation is $8+12h$ and the exact error is $6h^2+h^3$. For $h=0.01$, the error is $0.000601$, explicitly smaller than the linear change.

The continuous function $|x|$ has difference quotient $1$ for positive $h$ and $-1$ for negative $h$ at zero. It has no derivative there. The converse failure is therefore concrete: continuity alone does not supply a tangent.

Important properties

Differentiability implies continuity because both the linear increment and the remainder vanish. Product and chain rules follow by expanding increments and controlling their remainders; the composition rule multiplies local linear approximations.

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, the mean value theorem gives some $c\in(a,b)$ with $f(b)-f(a)=f'(c)(b-a)$. Its proof subtracts the secant line and applies the interior-extremum derivative condition. A derivative bounded in absolute value by $M$ therefore yields a Lipschitz estimate on an interval. Stationary points need not be extrema: $x^3$ has derivative zero at zero.

Connections and further reading

The fundamental theorem of calculus connects derivatives with accumulation. Weak derivatives extend differentiation through integration against test functions, allowing certain corners. Taylor, §4.1 supplies the local expansion and the mean value argument. In several variables the corresponding object is a linear map, requiring more than separate coordinate derivatives.