Answer
$f'\left( x \right) = - \frac{2}{{\sqrt {1 + 4{x^2}} }}$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {\sinh ^{ - 1}}\left( { - 2x} \right) \cr
& {\text{Differentiate}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {{{\sinh }^{ - 1}}\left( { - 2x} \right)} \right] \cr
& {\text{Use Derivatives of Inverse Hyperbolic Functions }} \cr
& \frac{d}{{dx}}\left[ {{{\sinh }^{ - 1}}u} \right] = \frac{1}{{\sqrt {1 + {u^2}} }}\frac{{du}}{{dx}},{\text{ let }}u = - 2x,{\text{ so}} \cr
& f'\left( x \right) = \frac{1}{{\sqrt {1 + {{\left( { - 2x} \right)}^2}} }}\frac{d}{{dx}}\left[ { - 2x} \right] \cr
& {\text{Compute the derivative and simplify}} \cr
& f'\left( x \right) = \frac{1}{{\sqrt {1 + {{\left( { - 2x} \right)}^2}} }}\left( { - 2} \right) \cr
& f'\left( x \right) = - \frac{2}{{\sqrt {1 + 4{x^2}} }} \cr} $$
