732. Find the derivative of the exponential-power function

$$
y=u^{v}(u>0) \text {, where } u=f(x), v=\varphi(x) \text {. }
$$
Solution. First, we take the logarithm of the given function (with base e): $\ln y=v \ln u$. Now we differentiate both sides of the equation using the rule for differentiating a composite function. We get

$$
\frac{y^{\prime}}{y}=v^{\prime} \ln u+v \frac{u^{\prime}}{u}
$$

from which

$$
y^{\prime}=y\left(v^{\prime} \ln u+v \frac{u^{\prime}}{u}\right)
$$

or

$$
y^{\prime}=u^{v}\left(v^{\prime} \ln u+\frac{v}{u} u^{\prime}\right)
$$