The definition of a Derivative of a Function

${\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {f(x+{\Delta }x)-f(x)}{{\Delta }x}}}$

Example

${\displaystyle f(x)=x^{2}}$
Use the limit definition with the given function

${\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {(x+{\Delta }x)^{2}-x^{2}}{{\Delta }x}}}$

${\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {(x^{2}+2x{\Delta }x+{\Delta }x^{2})-x^{2}}{{\Delta }x}}}$

${\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {2x{\Delta }x+{\Delta }x^{2}}{{\Delta }x}}}$

${\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {{\Delta }x(2x+{\Delta }x)}{{\Delta }x}}}$

${\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}(2x+{\Delta }x)}$

${\displaystyle f'(x)=2x+0}$

${\displaystyle f'(x)=2x}$