Answer
$$f'\left( x \right) = \frac{4}{{\sqrt {16{x^2} - 1} }}$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {\cosh ^{ - 1}}4x \cr
& {\text{find the derivative}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left( {{{\cosh }^{ - 1}}4x} \right) \cr
& {\text{use derivatives of the inverse hyperbolic functions}} \cr
& f'\left( x \right) = \frac{1}{{\sqrt {{{\left( {4x} \right)}^2} - 1} }}\frac{d}{{dx}}\left( {4x} \right) \cr
& f'\left( x \right) = \frac{1}{{\sqrt {{{\left( {4x} \right)}^2} - 1} }}\left( 4 \right) \cr
& {\text{simplify}} \cr
& f'\left( x \right) = \frac{4}{{\sqrt {16{x^2} - 1} }} \cr} $$
