Answer
$\frac{d^2y}{dx^2}[xcos(x^2)]= -6xsin(x^2)-4x^3cos(x^2)$
Work Step by Step
Chain Rule
$\frac{d}{dx}[f(g(x))] = f'(g(x)) \times g'(x)$
$y=xcos(x^2)$
Product Rule and Chain Rule:
$\frac{d}{dx}[xcos(x^2)] = cos(x^2) + x(2x(-sin(x^2))) = cos(x^2)-2x^2sin(x^2)$
Product Rule and Chain Rule:
$\frac{d^2}{dx^2}[cos(x^2)-2x^2sin(x^2)] = [-2xsin(x^2)]+[-4xsin(x^2)-2x^2(2x)cos(x^2)]=[-2xsin(x^2)]+[-4xsin(x^2)-4x^3cos(x^2)] = -6xsin(x^2)-4x^3cos(x^2)$
