6.21. Find the derivative $y_{x}^{\prime}(x)$ of the function $f(x, y(x))=$ $=\ln y(x)+\cot x^{2}-2 x=0$, given implicitly.
Solution.

$$
\begin{gathered}
\left(\ln y(x)+\cot x^{2}-2 x\right)_{x}^{\prime}=(0)_{x}^{\prime} \\
\frac{1}{y(x)} \cdot y_{x}^{\prime}(x)-\frac{2 x}{\sin ^{2} x^{2}}-2=0
\end{gathered}
$$

from here $y_{x}^{\prime}(x)=y(x) \cdot\left(2 x /\left(\sin ^{2} x^{2}\right)+2\right)$.