Definition of Derivatives

For a function $f(x)$, the derivative $f’(x)$ at a point $x$ is defined as the limit of the difference quotient as it approaches zero: \(f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}.\)

We say $f$ is differentiable at $x=c$, if the $f’(c)$ exists.

Theorem. If $f(x)$ is differentiable at $x=c$, then $f$ is continuous at $c$.

We will use the definition of derivatives to calculate the derivatives of functions if the functions are either basic functions or conditional functions.