Answer
$$ - 45{\sin ^4}x\cos x$$
Work Step by Step
$$\eqalign{
& y = - 9{\sin ^5}x \cr
& {\text{we can write the function as}} \cr
& y = - 9{\left( {\sin x} \right)^5} \cr
& {\text{differentiate with respect to }}x \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ { - 9{{\left( {\sin x} \right)}^5}} \right] \cr
& \frac{{dy}}{{dx}} = - 9\frac{d}{{dx}}\left[ {{{\left( {\sin x} \right)}^5}} \right] \cr
& {\text{use the general power rule for derivatives }}\frac{d}{{dx}}\left[ {{u^n}} \right] = n{u^{n - 1}}\frac{{du}}{{dx}}.{\text{ consider }}u = \cos x \cr
& then \cr
& \frac{{dy}}{{dx}} = - 9\left( 5 \right){\left( {\sin x} \right)^{5 - 1}}\frac{d}{{dx}}\left[ {\sin x} \right] \cr
& \frac{{dy}}{{dx}} = - 9\left( 5 \right){\left( {\sin x} \right)^4}\left( {\cos x} \right) \cr
& {\text{simplifying}} \cr
& \frac{{dy}}{{dx}} = - 45{\sin ^4}x\cos x \cr} $$
