reciprocal rule
Reciprocal rule
The reciprocal rule is written as \[\left( \frac{1}{f(x)} \right)^{\prime}=-\frac{f^{\prime}(x)}{(f(x))^{2}}\].
Proof
\begin{align*}
\left( \frac{1}{f(x)} \right)^{\prime}&=\lim_{h\to0}\frac{\left(\displaystyle \frac{1}{f(x+h)} \right)-\left( \displaystyle \frac{1}{f(x)} \right)}{h}\\
&=\lim_{h\to0}\frac{\left( \displaystyle \frac{f(x)-f(x+h)}{f(x+h)f(x)} \right)}{h}\\
&=\lim_{h\to0}\frac{f(x)-f(x+h)}{hf(x+h)f(x)}\\
&=\left( \lim_{h\to0}-\frac{f(x+h)-f(x)}{h} \right)\cdot \left( \lim_{h\to0}\frac{1}{f(x+h)}\cdot \frac{1}{f(x)} \right)\\
&=-f^{\prime}(x)\cdot \frac{1}{(f(x))^{2}}\\
&=-\frac{f^{\prime}(x)}{(f(x))^{2}}\\
\end{align*}