Now that we know the derivative of a log, we can combine it with the
chain rule:$$\frac{d}{dx}\Big( \ln(y)\Big)= \frac{1}{y} \frac{dy}{dx},$$ or
equivalently $$\frac{dy}{dx} = y \frac{d }{dx}\Big(\ln(y)\Big).$$ Sometimes it
is (much!) easier to take the derivative of $\ln(y)$ than of $y$. In those
cases, we can use the last equation to get $dy/dx$. This is
called logarithmic differentiation.
Example: Find the derivative of $y=x^x$.
Solution:
Take the log of both sides to get
$\ln(y) = x \ln(x)$. By the product rule, the derivative of
$x \ln(x)$ is $\ln(x) + 1$. So
$$ \frac{dy}{dx} = y \frac{d}{dx}\Big(\ln(y)\Big) = x^x (1+\ln(x)).$$