Example 3.27. Find $\frac{d z}{d t}$, if $z=\arcsin \left(x^{2}+y^{2}+t^{2}\right)$, where $x=2 t, y=4 t^{2}$.
Solution. The required derivative is found using formula (3.13). We find the necessary partial derivatives

$$
\begin{gathered}
\frac{\partial z}{\partial x}=\frac{2 x}{\sqrt{1-\left(x^{2}+y^{2}+t^{2}\right)^{2}}} ; \frac{\partial z}{\partial y}=\frac{2 y}{\sqrt{1-\left(x^{2}+y^{2}+t^{2}\right)^{2}}} \\
\frac{\partial z}{\partial t}=\frac{2 t}{\sqrt{1-\left(x^{2}+y^{2}+t^{2}\right)^{2}}} ; \frac{d x}{d t}=2 ; \frac{d y}{d t}=8 t
\end{gathered}
$$

Substituting into (3.13), we get:

$$
\frac{d z}{d t}=\frac{2 t+2 x \cdot 2+2 y \cdot 8 t}{\sqrt{1-\left(x^{2}+y^{2}+t^{2}\right)^{2}}}=\frac{2 t\left(1+4 t+32 t^{2}\right)}{\sqrt{1-\left(x^{2}+y^{2}+t^{2}\right)^{2}}}
$$