The definition of a Derivative of a Function
f ′ ( x ) = lim Δ x → 0 f ( x + Δ x ) − f ( x ) Δ x {\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {f(x+{\Delta }x)-f(x)}{{\Delta }x}}}
Example
f ( x ) = x 2 {\displaystyle f(x)=x^{2}} Use the limit definition with the given function
f ′ ( x ) = lim Δ x → 0 ( x + Δ x ) 2 − x 2 Δ x {\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {(x+{\Delta }x)^{2}-x^{2}}{{\Delta }x}}}
f ′ ( x ) = lim Δ x → 0 ( x 2 + 2 x Δ x + Δ x 2 ) − x 2 Δ x {\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {(x^{2}+2x{\Delta }x+{\Delta }x^{2})-x^{2}}{{\Delta }x}}}
f ′ ( x ) = lim Δ x → 0 2 x Δ x + Δ x 2 Δ x {\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {2x{\Delta }x+{\Delta }x^{2}}{{\Delta }x}}}
f ′ ( x ) = lim Δ x → 0 Δ x ( 2 x + Δ x ) Δ x {\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {{\Delta }x(2x+{\Delta }x)}{{\Delta }x}}}
f ′ ( x ) = lim Δ x → 0 ( 2 x + Δ x ) {\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}(2x+{\Delta }x)}
f ′ ( x ) = 2 x + 0 {\displaystyle f'(x)=2x+0}
f ′ ( x ) = 2 x {\displaystyle f'(x)=2x}
