1003. Find the total differentials of the functions:

$$
\text { 1) } z=5 x^{2} y^{3} ; \quad \text { 2) } z=\operatorname{arctg}\left(x^{2}+3 y\right)
$$
Solution. 1) First, we find the partial derivatives:

$$
\frac{\partial z}{\partial x}=10 x y^{3}, \frac{\partial z}{\partial y}=15 x^{2} y^{2}
$$

By formula (2) we get

$$
\begin{aligned}
& d z=10 x y^{3} d x+15 x^{2} y^{2} d y=5 x y^{2}(2 y d x+3 x d y) \\
& \text { 2) } \frac{\partial z}{\partial x}=\frac{\left(x^{2}+3 y\right)_{x}^{\prime}}{1+\left(x^{2}+3 y\right)^{2}}=\frac{2 x}{1+\left(x^{2}+3 y\right)^{2}}, \quad \frac{\partial z}{\partial y}=\frac{\left(x^{2}+3 y\right)_{y}^{\prime}}{1+\left(x^{2}+3 y\right)^{2}}= \\
& =\frac{3}{1+\left(x^{2}+3 y\right)^{2}} ; d z=\frac{1}{1+\left(x^{2}+3 y\right)^{2}}(2 x d x+3 d y)
\end{aligned}
$$