# Reciprocal Rule The *reciprocal rule* is a formula for the [[derivative]] of the *reciprocal of a function*. >[!Example] Reciprocal Rule > If $\displaystyle g(x)=\frac{1}{f(x)}$, then > $g'(x)=-\frac{f'(x)}{f(x)^{2}}$ > for *non-zero* $f(x)$. ## Proof A proof of the reciprocal rule uses the limit definition of the derivative. Let $\displaystyle g(x)=\frac{1}{f(x)}$ and $f$ be differentiable at $x$. $\begin{align*} g'(x)&=\lim_{h\rightarrow 0}\frac{g(x+h)-g(x)}{h} \\ &=\lim_{h\rightarrow 0}\frac{\displaystyle\frac{1}{f(x+h)}-\frac{1}{f(x)}}{h} \\ &=\lim_{h\rightarrow 0}\frac{f(x)-f(x+h)}{h\;f(x)f(x+h)} \\ &=\lim_{h\rightarrow 0}-\frac{f(x+h)-f(x)}{h}\frac{1}{f(x)f(x+h)} \\ &=\lim_{h\rightarrow 0}-\frac{f(x+h)-f(x)}{h}\lim_{h\rightarrow 0}\frac{1}{f(x)f(x+h)} \\ &=-f'(x)\frac{1}{f(x)^{2}} \\ \therefore g'(x)&=-\frac{f'(x)}{f(x)^{2}} \end{align*}$