Answer
$$\frac{d^2}{dx^2}\sin (x^2)=2\cos(x^2)-4x^2\sin (x^2).$$
Work Step by Step
Recall that $(\sin x)'=\cos x$.
Recall that $(\cos x)'=-\sin x$.
Recall that $(x^n)'=nx^{n-1}$
We have
$\frac{d}{dx}\sin (x^2)=\cos (x^2) (2x)=2x\cos(x^2)$
and hence
$\frac{d^2}{dx^2}\sin (x^2)=2\frac{d}{dx} x\cos(x^2)$
Use the product rule:
$=2\cos(x^2)-2x\sin (x^2)(2x)$
$=2\cos(x^2)-4x^2\sin (x^2)$
