Using the definition $f'(x)=\lim\limits_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$, find $f'(x)$ for $f(x)=\dfrac{1}{x}$.

Core Concept

The difference quotient simplifies to $\dfrac{\frac{1}{x+h}-\frac{1}{x}}{h}=\dfrac{-1}{x(x+h)}$. The limit as $h\to 0$ is $-\dfrac{1}{x^2}$.