Condition of the problem

Find the derivative.

$y=\frac{\operatorname{ch} x}{\sqrt{\operatorname{sh} 2 x}}$
## Solution

$$
\begin{aligned}
& y^{\prime}=\left(\frac{\cosh x}{\sqrt{\sinh 2 x}}\right)^{\prime}=\left(\frac{\cosh x}{\sqrt{2 \sinh x \cdot \cosh x}}\right)^{\prime}=\left(\sqrt{\frac{\cosh x}{2 \sinh x}}\right)^{\prime}= \\
& =\frac{1}{\sqrt{2}} \cdot\left(\sqrt{\frac{\cosh x}{\sinh x}}\right)^{\prime}=\frac{1}{\sqrt{2}} \cdot \frac{1}{2 \sqrt{\frac{\cosh x}{\sinh x}}} \cdot \frac{\sinh x \cdot \sinh x-\cosh x \cdot \cosh x}{\sinh^{2} x}= \\
& =-\frac{1}{2 \sqrt{2}} \cdot \sqrt{\frac{\sinh x}{\cosh x}} \cdot \frac{\cosh^{2} x-\sinh^{2} x}{\sinh^{2} x}=-\frac{1}{2 \sqrt{2 \cosh x \cdot \sinh x}} \cdot \frac{1}{\sinh x}= \\
& =-\frac{1}{2 \sinh x \sqrt{\sinh 2 x}}
\end{aligned}
$$

Kuznetsov Differentiation 11-13