Answer
$$ y''=32 \cos (8x+18).$$
Work Step by Step
Recall that $(\sin x)'=\cos x$.
Given $ y=\sin^2 (4x+9)$, then, by using the chain rule, we have
$$ y'=2\sin(4x+9) \cos (4x+9) (4)=8\sin(4x+9) \cos (4x+9) .$$
One can simplify $ y'$ using the fact that $2\sin x \cos x=\sin2x $ as follows
$$ y'=4\sin (8x+18)$$
and, by using the chain rule, the second derivative $ y''$ is given by
$$ y''=4\cos (8x+18) (8)=32 \cos (8x+18).$$
