Example 3. Calculate the value of the derivative of the implicit function $x y^{2}=4$ at the point $M(1,2)$.
Solution. First, let's find the derivative:

$$
x^{\prime} y^{2}+x 2 y y^{\prime}=0, y^{\prime}=-\frac{y}{2 x}
$$

Substituting the values \(x=1\), \(y=2\) into the right-hand side of the last equation, we get

$$
y^{\prime}=-\frac{2}{2 \cdot 1}=-1
$$