Answer
$$\frac{8}{{17}}$$
Work Step by Step
$$\eqalign{
& {\text{ }}f\left( x \right) = {\tan ^{ - 1}}\left( {4{x^2}} \right) \cr
& {\text{find the derivative}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {{{\tan }^{ - 1}}\left( {4{x^2}} \right)} \right] \cr
& {\text{use the formula }}\frac{d}{{dx}}\left[ {{{\tan }^{ - 1}}u} \right] = \frac{1}{{1 + {u^2}}}\frac{{du}}{{dx}}.{\text{ consider }}u = 4{x^2} \cr
& f'\left( x \right) = \frac{1}{{1 + {{\left( {4{x^2}} \right)}^2}}}\frac{d}{{dx}}\left[ {4{x^2}} \right] \cr
& {\text{solve the derivative and simplify}} \cr
& f'\left( x \right) = \frac{1}{{1 + 16{x^4}}}\left( {8x} \right) \cr
& f'\left( x \right) = \frac{{8x}}{{1 + 16{x^4}}} \cr
& \cr
& {\text{calculate }}f'\left( 1 \right) \cr
& f'\left( 1 \right) = \frac{{8\left( 1 \right)}}{{1 + 16{{\left( 1 \right)}^4}}} \cr
& f'\left( 1 \right) = \frac{8}{{17}} \cr} $$
