Answer
$$ y'= 5\tanh (x ) (\ln \cosh(x))^4$$
Work Step by Step
Recall that $(\ln x)'=\dfrac{1}{x}$
Recall that $(\cosh x)'=\sinh x$
Since $ y=(\ln \cosh(x))^5$, then the derivative, by using the chain rule, is given by
$$ y'=5(\ln \cosh(x))^4(\ln \cosh(x))'=5(\ln \cosh(x))^4(\frac{1}{\cosh x})(\cosh x)'\\
=5(\ln \cosh(x))^4\frac{\sinh x}{\cosh x}=5\tanh (x ) (\ln \cosh(x))^4.$$
