For x^x neither standard rule applies: the power rule needs a constant exponent, the exponential
rule a constant base. Taking logarithms of both sides turns the exponent into an ordinary factor, and
the problem becomes routine.
The mechanics
Write y = f^g. Then ln y = g ln f. Differentiating both sides gives y'/y = g' ln f + g f'/f, and
multiplying back by y finishes it. The two terms have a plain reading: the first is the change coming
from the exponent, the second from the base.
The other use
The trick is useful even when it is not required. A product of five factors needs four nested product rules, or one logarithm, one sum of five simple derivatives, and one multiplication back. For messy products and quotients it is frequently the shortest correct route.
See the logarithmic differentiation page for five worked cases,
including x^(1/x), whose maximum at x = e is the reason e^(1/e) beats every other x^(1/x).