Example 1. Find the derivative of the function $z=x^{2}+y^{2}$ at the point $M(3,1)$ in the direction of the point $M_{1}(0,5)$.
Solution. We have: $\overrightarrow{M M}_{1}=\{0-3,5-1\}=\{-3,4\},\left|\overrightarrow{M M}_{\mathrm{I}}\right|=$ $=\sqrt{(-3)^{2}+4^{2}}=5, \cos \alpha=-\frac{3}{5}, \sin \alpha=\frac{4}{5}$.

Let $\vec{l}=\left\{-\frac{3}{5}, \frac{4}{5}\right\}$ and find $\frac{\partial z}{\partial l}$.

We have: $z_{x}^{\prime}=2 x, z_{y}^{\prime}=2 y, z_{x}^{\prime}(3,1)=6, z_{y}^{\prime}(3,1)=2$.

Finally, $\frac{\partial z}{\partial l}=6\left(-\frac{3}{5}\right)+2 \cdot \frac{4}{5}=-2$. The negative sign of the derivative value $\frac{\partial z(3,1)}{\partial l}$ indicates that the function is decreasing in the direction of $\vec{l}$.