## Task Condition

Find the derivative.

$y=\left(x^{2}-1\right)^{\sin x}$
## Solution

$y=\left(x^{2}-1\right)^{\sin x}$

$\ln y=\ln \left(x^{2}-1\right)^{\sinh x}=\sinh x \cdot \ln \left(x^{2}-1\right)$

$(\ln y)^{\prime}=\frac{y^{\prime}}{y}=\cosh x \cdot \ln \left(x^{2}-1\right)+\sinh x \cdot \frac{1}{x^{2}-1} \cdot 2 x=$

$=\cosh x \cdot \ln \left(x^{2}-1\right)+\frac{2 x \cdot \sinh x}{x^{2}-1}$
$y^{\prime}=y \cdot\left(\cosh x \cdot \ln \left(x^{2}-1\right)+\frac{2 x \cdot \sinh x}{x^{2}-1}\right)=$

$=\left(x^{2}-1\right)^{\sinh x} \cdot\left(\cosh x \cdot \ln \left(x^{2}-1\right)+\frac{2 x \cdot \sinh x}{x^{2}-1}\right)$

## Problem Kuznetsov Differentiation 12-15