## Problem Statement

Find the derivative.

$$
y=\operatorname{arctan} \frac{\sqrt{\operatorname{sinh} 2 x}}{\operatorname{cosh} x-\operatorname{sinh} x}
$$
## Solution

$$
\begin{aligned}
& y^{\prime}=\left(\operatorname{arctg} \frac{\sqrt{\operatorname{sh} 2 x}}{\operatorname{ch} x-\operatorname{sh} x}\right)^{\prime}=\frac{1}{1+\left(\frac{\sqrt{\operatorname{sh} 2 x}}{\operatorname{ch} x-\operatorname{sh} x}\right)^{2}} \cdot\left(\frac{\sqrt{\operatorname{sh} 2 x}}{\operatorname{ch} x-\operatorname{sh} x}\right)^{\prime}= \\
& =\frac{(\operatorname{ch} x-\operatorname{sh} x)^{2}}{(\operatorname{ch} x-\operatorname{sh} x)^{2}+\operatorname{sh} 2 x} \cdot \frac{\frac{1}{2 \sqrt{\operatorname{sh} 2 x}} \cdot \operatorname{ch} 2 x \cdot 2 \cdot(\operatorname{ch} x-\operatorname{sh} x)-\sqrt{\operatorname{sh} 2 x} \cdot(\operatorname{sh} x-\operatorname{ch} x)}{(\operatorname{ch} x-\operatorname{sh} x)^{2}}= \\
& =\frac{\frac{\operatorname{ch} 2 x}{\sqrt{\operatorname{sh} 2 x}} \cdot(\operatorname{ch} x-\operatorname{sh} x)+\sqrt{\operatorname{sh} 2 x} \cdot(\operatorname{ch} x-\operatorname{sh} x)}{\operatorname{ch}^{2} x-2 \operatorname{ch} x \cdot \operatorname{sh} x+\operatorname{sh}^{2} x+2 \operatorname{sh} x \cdot \operatorname{ch} x}= \\
& =\frac{(\operatorname{ch} x-\operatorname{sh} x) \cdot\left(\frac{\operatorname{ch} 2 x}{\sqrt{\operatorname{sh} 2 x}}+\sqrt{\operatorname{sh} 2 x}\right)}{\operatorname{ch}^{2} x+\operatorname{sh}^{2} x}=\frac{(\operatorname{ch} x-\operatorname{sh} x) \cdot(\operatorname{ch} 2 x+\operatorname{sh} 2 x)}{\sqrt{\operatorname{sh} 2 x} \cdot\left(\operatorname{ch}^{2} x+\operatorname{sh}^{2} x\right)}= \\
& =\frac{(\operatorname{ch} x-\operatorname{sh} x) \cdot\left(\operatorname{ch}^{2} x+\operatorname{sh}^{2} x+2 \operatorname{sh} x \cdot \operatorname{ch} x\right)}{\sqrt{\operatorname{sh} 2 x} \cdot \operatorname{ch} 2 x}= \\
& =\frac{(\operatorname{ch} x-\operatorname{sh} x) \cdot(\operatorname{ch} x+\operatorname{sh} x)^{2}}{\sqrt{\operatorname{sh} 2 x} \cdot \operatorname{ch} 2 x}=\frac{\left(\operatorname{ch}^{2} x-\operatorname{sh}^{2} x\right) \cdot(\operatorname{ch} x+\operatorname{sh} x)}{\sqrt{\operatorname{sh} 2 x} \cdot \operatorname{ch} 2 x}= \\
& =\frac{1 \cdot(\operatorname{ch} x+\operatorname{sh} x)}{\sqrt{\operatorname{sh} 2 x} \cdot \operatorname{ch} 2 x}=\frac{\operatorname{ch} x+\operatorname{sh} x}{\sqrt{\operatorname{sh} 2 x} \cdot \operatorname{ch} 2 x}
\end{aligned}
$$

## Problem Kuznetsov Differentiation 11-9