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The definition of the derivative

The derivative is the limit of the slope of the secant as the distance between the points tends to zero. The difference quotient is the change in $y$ divided by the change in $x$ . For a polynomial you can cancel $h$ before you let $h$ tend to zero. A small but nonzero step gives a value close to the derivative, and that is a practical check of the calculation.

Try it yourself

The derivative as a slope

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

the definition

\frac{(x+h)^2-x^2}{h} = 2x + h

for the square function

Symbols

$h$	the step
$f'(x)$	the derivative

Example

For $f(x) = x^2$ and the step $0.1$ from $x = 3$ : $\dfrac{3.1^2 - 9}{0.1} = 6.1$ . The formula $2x + h$ gives the same, $6.1$ .

The derivative is the limit as the step tends to zero, not the fraction for one particular step.

Practise limits and continuity for free →

← Continuity at a joint · The derivative of the exponential →

Part of Mathematics R1: Limits and continuity.