Definition

Derivative

\frac{df}{d x} = Δ x \to 0 lim \frac{f ( x + Δ x ) - f ( x )}{Δ x}

Statement

The limiting value of the average rate of change of f, as the interval over which it is measured shrinks to zero.

Symbols and units

$f, x$	the function and its variable	—
$Δ x$	small change in x	—

When it applies

Functions that are continuous and smooth at the point in question.

Don't use when

Not defined at a discontinuity, a sharp corner, or a vertical tangent — the limit does not exist there.

Derivation

Compute the average rate of change over a finite interval Δx:

\frac{Δ f}{Δ x} = \frac{f ( x + Δ x ) - f ( x )}{Δ x}

Shrink the interval to zero; the ratio approaches the instantaneous rate:

\frac{df}{d x} = Δ x \to 0 lim \frac{f ( x + Δ x ) - f ( x )}{Δ x}

Worked examples

Velocity from position

x(t) = 5t² (metres, t in seconds):

v = \frac{d x}{d t} = 10 t

At t = 2 s:

v = 20 m/s

Watch out for

A derivative exists only where the function is smooth — it is undefined at a kink, a jump, or a vertical tangent.

Connects to