Derivative of x^2 sin(x) via the product rule

Identify the two factors: $u(x) = x^2$ and $v(x) = \sin(x)$. By the power rule, $u'(x) = 2x$, and by the memorized trigonometric derivative rule, $v'(x) = \cos(x)$.

Apply the product rule formula $g'(x) = u'(x)v(x) + u(x)v'(x)$, substituting in the pieces found above: $g'(x) = 2x\sin(x) + x^2\cos(x)$. This is the fully simplified derivative.

The derivative of cosine of x is negative sine of x. This is one of the six memorized trigonometric derivatives, and the negative sign is a common point of confusion. In part a, you used the derivative of sine of x, which is positive cosine of x, showing how the two are related but differ in sign.