JAMB 2000 · UME · Q36

If $y = 2x\cos2x - \sin2x$ , find $\frac{dy}{dx}$ when $x = \frac\pi4$ .

Worked solution (try it first)

Product rule on $2x\cos2x$ : $2\cos2x + 2x(-2\sin2x) = 2\cos2x - 4x\sin2x$ .
Chain rule on $\sin2x$ : $2\cos2x$ .
Subtract it: $\frac{dy}{dx} = -4x\sin2x$ .
At $x = \frac\pi4$ , $\sin2x = \sin\frac\pi2 = 1$ , so $\frac{dy}{dx} = -4 \times \frac\pi4$
$= -\pi$ , option B.