Differential¶

Definition¶

Let $f$ be a function differentiable at a point $x\text{.}$ The differential of $f$ at $x$ is defined as

\mathrm{d}f(x) = f'(x)\,\mathrm{d}x,

where $\mathrm{d}x$ denotes the differential of the independent variable $x\text{.}$

Interpretation¶

The differential gives the linear part of the change of $f$ near a point: for an increment $\mathrm{d}x$ of the input, $\mathrm{d}f=f'(x)\,\mathrm{d}x$ is the change that the tangent line predicts. It is the bridge between the derivative, a single number, and linear approximation, a statement about changes.

A curve and its tangent at x0; over a step dx, the rise dy of the tangent is compared with the actual rise delta y of the curve, which is close for small dx.

$\mathrm{d}y=f'(x_0)\,\mathrm{d}x$ is the rise of the tangent; $\Delta y=f(x_0+\mathrm{d}x)-f(x_0)$ is the rise of $f$ itself. Here the curve bends upwards, so $\Delta y$ is larger than $\mathrm{d}y$ . For small $\mathrm{d}x$ the two are close: the gap $\Delta y-\mathrm{d}y$ is small even compared with $\mathrm{d}x$ .

The tangent's rise $\mathrm{d}f=f'(x_0)\,\mathrm{d}x$ approximates the actual change $\Delta f$ . Shrink $\mathrm{d}x$ and compare the error with $\mathrm{d}x$ itself.

At a fixed point $x$ , the differential is a linear function of the increment: doubling $\mathrm{d}x$ doubles $\mathrm{d}f$ . Dividing by $\mathrm{d}x$ gives back $\frac{\mathrm{d}f}{\mathrm{d}x}=f'(x)$ $f^{'} (x)$ , which is one way to read the Leibniz notation for the derivative.

A common slip is to treat $\mathrm{d}f$ as the exact change. At a given point, $\mathrm{d}f$ and $\Delta f$ agree for every increment only when $f$ is a linear function $ax+b$ $b$ ; otherwise $\mathrm{d}f$ is an approximation whose quality depends on how small $\mathrm{d}x$ is.

The notation returns later in integration by substitution, where $g'(x)\,\mathrm{d}x$ is replaced by $\mathrm{d}t$ for $t=g(x)$ $g (x)$ .

Key Results¶

The differential depends linearly on the increment $\mathrm{d}x\text{.}$
Differentiability guarantees the existence of a differential.
Linear approximation. If $f$ is differentiable at $x_0\text{,}$ then for $x$ close to $x_0$ its tangent line gives

$f(x)\approx f(x_0)+f'(x_0)(x-x_0).$

This tangent-line (linear) approximation is the form in which the differential is used in Math I: Calculus.

Examples¶

For $f(x)=x^2\text{,}$ $\mathrm{d}f(x)=2x\,\mathrm{d}x\text{.}$
For $f(x)=\sin x\text{,}$ $\mathrm{d}f(x)=\cos x\,\mathrm{d}x\text{.}$
For $f(x)=x^3$ we have $f'(1)=3\text{,}$ so $f(1.001)\approx f(1)+f'(1)\times 0.001=1.003\text{,}$ while the exact value is $1.003003001\text{.}$

Prerequisites¶

Derivative and Differentiability

Sources¶

Math I: Calculus · Lecture 6

Aliases: differential of a function Keywords: linear approximation Symbols: delta

Last updated: 20 October 2026 03:02