Answer
$f'\left( x \right) = {\sec ^2}x - 4\cos x$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \tan x - 4\sin x \cr
& {\text{Differentiating}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {\tan x - 4\sin x} \right] \cr
& {\text{Use the difference rule for derivatives }} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {\tan x} \right] - \frac{d}{{dx}}\left[ {4\sin x} \right] \cr
& {\text{Pull out the constant 4}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {\tan x} \right] - 4\frac{d}{{dx}}\left[ {\sin x} \right] \cr
& {\text{Use the Derivatives of Trigonometric Functions }} \cr
& f'\left( x \right) = {\sec ^2}x - 4\left( {\cos x} \right) \cr
& {\text{Simplify}} \cr
& f'\left( x \right) = {\sec ^2}x - 4\cos x \cr} $$
