Answer
$f'\left( x \right) = 3\sinh 3x$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \cosh 3x \cr
& {\text{Differentiate}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {\cosh 3x} \right] \cr
& {\text{Use the Derivatives of Hyperbolic Functions }} \cr
& \frac{d}{{dx}}\left[ {\cos u} \right] = \sinh u\frac{{du}}{{dx}},{\text{ let }}u = 3x,{\text{ so}} \cr
& f'\left( x \right) = \sinh 3x\frac{d}{{dx}}\left[ {3x} \right] \cr
& {\text{Compute the derivative and simplify}} \cr
& f'\left( x \right) = 3\sinh 3x \cr} $$
