derivative

The instantaneous rate of change of a function — the slope of its tangent line at a point.

The derivative of $f$ at $x$ is $f'(x) = \displaystyle\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}$ , when this limit exists. It gives the slope of the tangent line at $x$ and the instantaneous rate of change of $f$ there.

Examples

For $f(x)=x^2$ , $f''(x)=2x$ , so the slope of the tangent line at $x=3$ is $6$ .