Derivative/R/Squaring/Directly/Example

We consider the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2}.$

The difference quotient for ${}a$ and ${}a+h$ is

${}{\frac {f(a+h)-f(a)}{h}}={\frac {(a+h)^{2}-a^{2}}{h}}={\frac {a^{2}+2ah+h^{2}-a^{2}}{h}}={\frac {2ah+h^{2}}{h}}=2a+h\,.$

The limit of this, as ${}h$ tends to ${}0$, is ${}2a$. The derivative of ${}f$ in ${}a$ is therefore ${}f'(a)=2a$.