All courses › Foundations of Mathematics › The derivative

The derivative

The derivative is the instantaneous rate of change: the slope of the tangent at one point. You get it by letting the interval in the average rate of change shrink towards zero. A positive derivative means the graph rises, negative that it falls.

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

the definition

y - f(a) = f'(a)(x - a)

tangent at the point a

Symbols

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

Example

$f(x) = x^2$ : $f'(x) = 2x$ , so at $x = 3$ the slope is 6.

Tangent: $y = 9 + 6(x - 3) = 6x - 9$ .

Where the derivative is zero the graph has a horizontal tangent: often a maximum or minimum.

Practise introduction to derivatives for free →

← Average rate of change · Differentiation rules →

Part of Foundations of Mathematics: Introduction to derivatives.