733. Find $y^{\prime}$, if $y=x^{x}$.
Solution. $\ln y=x \ln x$.

The derivative of the left side of the equation is found using the rule for differentiating a composite function, and the derivative of the right side is found using the rule for differentiating a product:

$$
\frac{y^{\prime}}{y}=x^{\prime} \ln x+x(\ln x)^{\prime}
$$

or

$$
\frac{y^{\prime}}{y}=\ln x+x \frac{1}{x}
$$

from which

$$
y^{\prime}=y(1+\ln x)
$$

or

$$
y^{\prime}=x^{x}(1+\ln x)
$$