## Task Condition

Find the derivative.

$y=8 \sin (\operatorname{ctg} 3)+\frac{1}{5} \cdot \frac{\sin ^{2} 5 x}{\cos 10 x}$
Solution

$y^{\prime}=\left(8 \sin (\operatorname{ctg} 3)+\frac{1}{5} \cdot \frac{\sin ^{2} 5 x}{\cos 10 x}\right)^{\prime}=0+\frac{1}{5} \cdot\left(\frac{\sin ^{2} 5 x}{\cos 10 x}\right)^{\prime}=$

$=\frac{1}{5} \cdot \frac{2 \sin 5 x \cdot \cos 5 x \cdot 5 \cdot \cos 10 x-\sin ^{2} 5 x \cdot(-\sin 10 x) \cdot 10}{\cos ^{2} 10 x}=$

$=\frac{\sin 10 x \cdot \cos 10 x+2 \sin ^{2} 5 x \cdot \sin 10 x}{\cos ^{2} 10 x}=\frac{\sin 10 x \cdot\left(\cos 10 x+2 \sin ^{2} 5 x\right)}{\cos ^{2} 10 x}=$

$=\frac{\sin 10 x \cdot\left(1-2 \sin ^{2} 5 x+2 \sin ^{2} 5 x\right)}{\cos ^{2} 10 x}=\frac{\sin 10 x}{\cos ^{2} 10 x}=\frac{\operatorname{tg} 10 x}{\cos 10 x}$

Problem Kuznetsov Differentiation 9-13