Example 1 (to $4^{\circ}$). Find $y^{\prime}$ if $10 x^{3}+4 x^{2} y+y^{2}=0$. Also find $y^{\prime}$ at $x=-2, y=4$.
Solution. Assume that $y=y(x)$. Differentiate the left-hand side of the given equation as a composite function, set the obtained expression to zero, and find $y^{\prime}$:

$$
30 x^{2}+8 x y+4 x^{2} y^{\prime}+2 y y^{\prime}=0
$$

From this, $y^{\prime}=\frac{-15 x^{2}-4 x y}{2 x^{2}+y}$. Substituting $x=-2, y=4$, we get $y^{\prime}(-2)=-\frac{7}{3}$