Example 3.15. Find the partial derivatives of the function

$$
f(x, y, z)=\left\{\begin{array}{lr}
x^{4}+\frac{2 x y z}{y^{2}+z^{2}}, & y^{2}+z^{2} \neq 0 \\
x^{4}, & y^{2}+z^{2} = 0
\end{array}\right.
$$

at the point $A(1,0,0)$.
Solution. Consider the function $f(x, 0,0)=x^{4}$. Let's find the derivative of this function: $f_{x}^{\prime}(x, 0,0)=4 x^{3}$ and $f_{x}^{\prime}(1,0,0)=4$. Next, consider the functions $f(1, y, 0)=1$ and $f(1,0, z)=1$. We have: $f_{x}^{\prime}(1, y, 0)=0, f_{z}^{\prime}(1,0, z)=0$ and, in particular, $f_{y}^{\prime}(1,0,0)=0$ and $f_{z}^{\prime}(1,0,0)=0$. Thus, the partial derivatives of the function at point $A(1,0,0)$ exist, but as shown in Example 3.11, the function is discontinuous at point $A(1,0,0)$. Therefore, the existence of partial derivatives at a point does not guarantee the continuity of the function at that point in general.

Remark. The above definition of partial derivatives usually applies to interior points of the domain of a function of several variables and, in general, is not suitable for boundary points of this domain, as it is not always possible to compute the partial increments of the function. Therefore, in the boundary points of the domain of functions of several variables, partial derivatives are defined as the limit values of these derivatives.

From the definition of partial derivatives as ordinary derivatives under the condition of fixing the values of all variables except one, for which the derivative is taken, it follows, in particular, that the geometric meaning of partial derivatives of a function of two variables $z=f(x, y)$ at a point $P_{0}\left(x_{0}, y_{0}\right)$ is analogous to the geometric meaning of the derivative of a function of one variable: $f_{x}^{\prime}(x, y)$ is the tangent of the angle of inclination to the $O X$ axis (relative to this axis) of the tangent at the point $M_{0}\left(x_{0}, y_{0}, f\left(x_{0}, y_{0}\right)\right)$ to the section of the surface $z=f(x, y)$ by the plane $y=y_{0}$.

Partial derivatives of higher orders at point $P_{0}$ are defined by transitioning from partial derivatives of order $r-1$ to partial derivatives of order $r$ according to the following relationship:

$$
\frac{\partial}{\partial x_{i_{r}}}\left(\frac{\partial^{r-1} f}{\partial x_{i_{r-1}} \ldots \partial x_{i_{1}}}\right)=\frac{\partial^{r} f}{\partial x_{i_{r}}} \partial x_{i_{r-1}} \ldots \partial x_{i_{1}} \equiv f_{x_{1_{1}} \ldots x_{r_{r-1} x_{r_{r}}}}^{(r)}
$$