Limit Definition of the Derivative

The derivative of a function at a point can be defined using limits, without any differentiation rules.

Statement

For a function $f(x)$, the derivative at $x$ is

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$$

This expression measures the instantaneous rate of change, or the slope of $f$ at $x$.