## Task Condition

Find the derivative.

$y=(x \sin x)^{8 \ln (x \sin x)}$
## Solution

$y=(x \sin x)^{8 \ln (x \sin x)}$

$\ln y=\ln \left((x \sin x)^{8 \ln (x \sin x)}\right)=8 \ln (x \sin x) \cdot \ln (x \sin x)=8 \ln ^{2}(x \sin x)$

$\frac{y^{\prime}}{y}=\left(8 \ln ^{2}(x \sin x)\right)^{\prime}=16 \ln (x \sin x) \cdot \frac{1}{x \sin x} \cdot(\sin x+x \cos x)=$

$=\frac{16 \ln (x \sin x) \cdot(1+x \cdot \operatorname{ctg} x)}{x}$

$y^{\prime}=y \cdot \frac{16 \ln (x \sin x) \cdot(1+x \cdot \operatorname{ctg} x)}{x}=$

$=(x \sin x)^{8 \ln (x \sin x)} \cdot \frac{16 \ln (x \sin x) \cdot(1+x \cdot \operatorname{ctg} x)}{x}=$

$=\frac{16(x \sin x)^{8 \ln (x \sin x)} \cdot \ln (x \sin x) \cdot(1+x \cdot \operatorname{ctg} x)}{x}$

## Problem Kuznetsov Differentiation 12-11