## Task Condition

Find the derivative.

$$
y=\frac{1}{3} \cdot \cos \left(\tan \frac{1}{2}\right)+\frac{1}{10} \cdot \frac{\sin ^{2} 10 x}{\cos 20 x}
$$
## Solution

$$
\begin{aligned}
& y^{\prime}=\left(\frac{1}{3} \cdot \cos \left(\operatorname{tg} \frac{1}{2}\right)+\frac{1}{10} \cdot \frac{\sin ^{2} 10 x}{\cos 20 x}\right)^{\prime}=0+\frac{1}{10} \cdot\left(\frac{\sin ^{2} 10 x}{\cos 20 x}\right)^{\prime}= \\
& =\frac{1}{10} \cdot \frac{2 \sin 10 x \cdot \cos 10 x \cdot 10 \cdot \cos 20 x-\sin ^{2} 10 x \cdot(-\sin 20 x) \cdot 20}{\cos ^{2} 20 x}= \\
& =\frac{\sin 20 x \cdot \cos 20 x+2 \sin ^{2} 10 x \cdot \sin 20 x}{\cos ^{2} 20 x}= \\
& =\frac{\sin 20 x \cdot\left(\cos 20 x+2 \sin ^{2} 10 x\right)}{\cos ^{2} 20 x}= \\
& =\frac{\sin 20 x \cdot\left(1-2 \sin ^{2} 10 x+2 \sin ^{2} 10 x\right)}{\cos ^{2} 20 x}= \\
& =\frac{\sin 20 x}{\cos ^{2} 20 x}=\frac{\operatorname{tg} 20 x}{\cos 20 x}
\end{aligned}
$$

Kuznetsov Differentiation 9-11