Example 9. $y=\frac{\cos ^{2} 2 x}{\tan x^{2}}$.
Solution. Differentiate the quotient of two complex functions:

$$
\begin{aligned}
y^{\prime}= & \frac{\left(\cos ^{2} 2 x\right)^{\prime} \cdot \operatorname{tg} x^{2}-\left(\operatorname{tg} x^{2}\right)^{\prime} \cdot \cos ^{2} 2 x}{\left(\operatorname{tg} x^{2}\right)^{2}}= \\
& =\frac{2 \cos 2 x \cdot(-\sin 2 x) \cdot 2 \operatorname{tg} x^{2}-\frac{2 x \cdot \cos ^{2} 2 x}{\cos ^{2} x^{2}}}{\operatorname{tg}^{2} x^{2}}=\frac{-\sin 4 x \cdot \sin 2 x^{2}-2 x \cos ^{2} 2 x}{\sin ^{2} x^{2}}
\end{aligned}
$$