Derivative

A derivative is the number $f'(a)$ defined for a function $f:I\to\mathbb{R}$ at a point $a\in I$ by $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$, provided this limit exists.

This is a special instance of a limit at a point applied to the difference quotient. Existence of the derivative is the basic notion behind differentiability in one variable , and it implies continuity via differentiability implies continuity .

Examples:

• For $f(x)=x^2$, the derivative exists everywhere and $f'(x)=2x$.
• For $f(x)=|x|$, the derivative does not exist at $a=0$.