Example 3.14. Find the partial derivatives of the function

$$
u=\left(x^{2}+y^{2}+z^{2}\right)^{\sin (x y z)}
$$
Solution. In this case, both the base and the exponent depend on independent variables. One way to find partial derivatives in this case is to logarithmize both sides of the equation defining the function (i.e., the same technique used when finding the derivative of a single-variable function in a similar situation). We have

$$
\ln u=\ln \left(x^{2}+y^{2}+z^{2}\right)^{\sin (x y z)}=\sin (x y z) \ln \left(x^{2}+y^{2}+z^{2}\right)
$$

Thus, $\ln u=\sin (x y z) \ln \left(x^{2}+y^{2}+z^{2}\right)$. Differentiating both sides of the obtained equation sequentially with respect to $x, y, z$, we will find the partial derivatives. It is necessary to keep in mind that the function $\ln u$, written on the left side of the obtained equation, represents a composite function. Therefore, its partial derivative with respect to any of the variables has the form $(\ln u)^{\prime}=\frac{u^{\prime}}{u}$. Taking this into account, we get

$$
\frac{u_{x}^{\prime}}{u}=y z \cos (x y z) \ln \left(x^{2}+y^{2}+z^{2}\right)+\sin (x y z) \frac{2 x}{x^{2}+y^{2}+z^{2}}
$$

From which

$$
u_{x}^{\prime}=u\left(y z \cos (x y z) \ln \left(x^{2}+y^{2}+z^{2}\right)+\frac{2 x \sin (x y z)}{x^{2}+y^{2}+z^{2}}\right)
$$

Or

$$
\begin{aligned}
u_{x}^{\prime}=\left(x^{2}\right. & \left.+y^{2}+z^{2}\right)^{\sin (x y z)} \times \\
& \times\left(y z \cos (x y z) \ln \left(x^{2}+y^{2}+z^{2}\right)+\frac{2 x \sin (x y z)}{x^{2}+y^{2}+z^{2}}\right)
\end{aligned}
$$

Similarly, the partial derivatives with respect to the variables $y$ and $z$ are found:

$$
\begin{aligned}
\frac{u_{y}^{\prime}}{u}= & x z \cos (x y z) \ln \left(x^{2}+y^{2}+z^{2}\right)+\frac{2 y \sin (x y z)}{x^{2}+y^{2}+z^{2}} \Rightarrow \\
\Rightarrow u_{y}^{\prime}= & \left(x^{2}+y^{2}+z^{2}\right)^{\sin (x y z)} \times \\
& \times\left(x z \cos (x y z) \ln \left(x^{2}+y^{2}+z^{2}\right)+\frac{2 y \sin (x y z)}{x^{2}+y^{2}+z^{2}}\right) \\
\frac{u_{z}^{\prime}}{u}= & x y \cos (x y z) \ln \left(x^{2}+y^{2}+z^{2}\right)+\frac{2 z \sin (x y z)}{x^{2}+y^{2}+z^{2}} \Rightarrow \\
\Rightarrow u_{z}^{\prime}= & \left(x^{2}+y^{2}+z^{2}\right)^{\sin (x y z)} \times \\
& \times\left(x y \cos (x y z) \ln \left(x^{2}+y^{2}+z^{2}\right)+\frac{2 z \sin (x y z)}{x^{2}+y^{2}+z^{2}}\right)
\end{aligned}
$$