JAMB 1992 · UME · Q37

If $y = x\sin x$ , find $\dfrac{d^2y}{dx^2}$ .

Worked solution (try it first)

Product rule, with $u = x$ and $v = \sin x$ : $\frac{dy}{dx} = 1 \cdot \sin x + x\cos x$
$= \sin x + x\cos x$ .
Differentiate again.
$\sin x$ gives $\cos x$ , and by the product rule $x\cos x$ gives $\cos x - x\sin x$ .
Add them: $\frac{d^2y}{dx^2} = 2\cos x - x\sin x$ , option A.