## Task Condition

Find the differential $d y$.

$$
y=x(\sin (\ln x)-\cos (\ln x))
$$
## Solution

$$
\begin{aligned}
& d y=y^{\prime} \cdot d x=(x(\sin (\ln x)-\cos (\ln x)))^{\prime} d x= \\
& =\left((\sin (\ln x)-\cos (\ln x))+x(\sin (\ln x)-\cos (\ln x))^{\prime}\right) d x= \\
& =\left(\sin (\ln x)-\cos (\ln x)+x\left((\sin (\ln x))^{\prime}-(\cos (\ln x))^{\prime}\right)\right) d x= \\
& =\left(\sin (\ln x)-\cos (\ln x)+x\left(\cos (\ln x) \cdot(\ln x)^{\prime}-\left(-\sin (\ln x) \cdot(\ln x)^{\prime}\right)\right)\right) d x= \\
& =\left(\sin (\ln x)-\cos (\ln x)+x\left(\cos (\ln x) \cdot \frac{1}{x}+\sin (\ln x) \cdot \frac{1}{x}\right)\right) d x= \\
& =(\sin (\ln x)-\cos (\ln x)+\cos (\ln x)+\sin (\ln x)) d x= \\
& =2 \sin (\ln x) \cdot d x
\end{aligned}
$$

## Problem Kuznetsov Differentiation 4-25