6.20. Find the derivative $y_{x}^{\prime}(x)$ of the function $f(x, y(x))=$ $=y^{2}(x)+x^{2}-1=0$, given implicitly.
S o l u t i o n. We will find the derivative of both sides of the equality. We have $\left(y^{2}(x)+x^{2}-1\right)_{x}^{\prime}=(0)_{x}^{\prime}, 2 \cdot y(x) \cdot y_{x}^{\prime}(x)+$ $+2 \cdot x=0$. Here, the expression $y^{2}(x)$ was considered as a complex two-link function: squaring - the first outer link, $y(x)$ - the second inner link. Hence, $y_{x}^{\prime}(x)=-x /(y(x))$.