6.22. Find the derivative $y_{x}^{\prime}(x)$ of the function

$$
f(x, y(x))=e^{y^{2}(x)}-\arcsin y(x)-x^{3}-1=0
$$

given implicitly.
Solution. $\left(e^{y^{2}(x)}-\arcsin y(x)-x^{3}-1\right)_{x}^{\prime}=(0)_{x}^{\prime}$,

$$
2 y(x) \cdot y_{x}^{\prime}(x) \cdot e^{y^{2}(x)}-\frac{y_{x}^{\prime}(x)}{\sqrt{1-y^{2}(x)}}-3 x^{2}=0,
$$

from here

$$
y_{x}^{\prime}(x)=\frac{3 x^{2}}{2 \cdot y(x) \cdot e^{y^{2}(x)}-\frac{1}{\sqrt{1-y^{2}(x)}}}
$$

## CALCULATION OF THE DERIVATIVE OF A FUNCTION GIVEN PARAMETRICALLY