The definition

If $f:D\rightarrow(-\infty,\infty)$ and the limit as h approaches of $${\frac{f(a+h)-f(a)}{h}}$$ exists, then we call the value of this limit the derivative of f at a, and we denote it by f'(a). We will write

$$f'(a) = \lim_{h\rightarrow 0}\frac{f(a+h) - f(a)}{h}$$

For example, suppose $f:(-\infty,\infty)\rightarrow(-\infty,\infty)$ and f(x) = x². Since

$$\lim_{h\rightarrow 0}\frac{(a+h)^2 - a^2}{h} = 2a$$

we know that f'(a) = 2a.