Using the definition $f'(x)=\lim\limits_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$, find $f'(x)$ for $f(x)=x^2$.

Core Concept

Expand $f(x+h)=(x+h)^2=x^2+2xh+h^2$, so the difference quotient is $\dfrac{2xh+h^2}{h}=2x+h$. The limit as $h\to 0$ is $2x$.