Question 34

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Jambmaths question: 

If $y=2x\cos 2x-\sin 2x$, find $\frac{dy}{dx}$when $x=\tfrac{\pi }{4}$ 

Option A: 

π

Option B: 

π

Option C: 

$\tfrac{\pi }{2}$

Option D: 

$-\tfrac{\pi }{2}$

Jamb Maths Solution: 

$\begin{align}  & y=2x\cos 2x-\sin 2x \\ & \frac{dy}{dx}=\underbrace{2x\frac{d}{dx}(\cos 2x)+\cos 2x\frac{d}{dx}(2x)}_{product\text{ }rule}-\frac{d}{dx}(\sin 2x) \\ & \frac{dy}{dx}=2x(-2\sin 2x)+\cos 2x(2)-2\cos 2x \\ & \frac{dy}{dx}=-4x\sin 2x+2\cos 2x-2\cos 2x \\ & \frac{dy}{dx}=-4x\sin 2x \\ & \text{At }x=\frac{\pi }{4}radian={{45}^{o}} \\ & \frac{dy}{dx}\left| _{x=\tfrac{\pi }{4}} \right.=-4(\tfrac{\pi }{4})\sin 2(\tfrac{\pi }{4}) \\ & \frac{dy}{dx}\left| _{x=\tfrac{\pi }{4}} \right.=-\pi \sin (\tfrac{\pi }{2})\text{(Note: sin}\tfrac{\pi }{2}=1) \\ & \frac{dy}{dx}\left| _{x=\tfrac{\pi }{4}} \right.=-\pi \text{    } \\\end{align}$

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