Derivative

Let $f:E\to\mathbb{R}$ (or $\mathbb{C}$) with $E\subseteq\mathbb{R}$ and let $a\in E$ be a limit point of $E$. If the limit

exists (this is the difference quotient ), it is called the derivative of $f$ at $a$ and is denoted $f'(a)$.

Equivalently, $f$ is differentiable at $a$ iff there exists a number $L$ such that

and then $L=f'(a)$.

Examples:

• If $f(x)=x^n$ with $n\in\mathbb{N}$, then $f'(a)=n a^{n-1}$.
• If $f(x)=e^x$, then $f'(a)=e^a$.
• If $f(x)=|x|$, then $f'(0)$ does not exist.