Question

Differentiate the following $w.r.t.x$.
$(x\sin x+\cos x)(x\cos x-\sin x)$

Answer 1

$\begin{array}{c}y=\left(x\sin x+\cos x\right)\left(x\cos x-\sin x\right)\\ \frac{dy}{dx}=\left(\sin x+x\cos x-\sin x\right)\left(x\cos x-\sin x\right)+\left(x\sin x+\cos x\right)\left(\cos x-x\sin x-\cos x\right)\\ =x\cos x\left(x\cos x-\sin x\right)-x\sin x\left(x\sin x+\cos x\right)\\ =x^2\left({\cos }^{2}x-{\sin }^{2}x\right)-2x\sin x\cos x\\ \therefore \frac{dy}{dx}=x^2\cos 2x-x\sin 2x\end{array}$