Answer
$$\frac{{dy}}{{dx}} = - 4{\cos ^3}x\sin x$$
Work Step by Step
$$\eqalign{
& y = {\cos ^4}x \cr
& {\text{we can write the function as}} \cr
& y = {\left( {\cos x} \right)^4} \cr
& {\text{differentiate with respect to }}x \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {{{\left( {\cos x} \right)}^4}} \right] \cr
& {\text{use the general power rule for derivatives }}\frac{d}{{dx}}\left[ {{u^n}} \right] = n{u^{n - 1}}\frac{{du}}{{dx}}.{\text{ consider }}u = \cos x \cr
& then \cr
& \frac{{dy}}{{dx}} = 4{\left( {\cos x} \right)^{4 - 1}}\frac{d}{{dx}}\left[ {\cos x} \right] \cr
& \frac{{dy}}{{dx}} = 4{\left( {\cos x} \right)^3}\left( { - \sin x} \right) \cr
& {\text{simplifying}} \cr
& \frac{{dy}}{{dx}} = - 4{\cos ^3}x\sin x \cr} $$
