Question

$\frac{sin2x}{aco{s}^{2}x+bsi{n}^{2}x}$

Answer 1

Let

$y=\frac{\sin 2x}{a{\cos }^{2}x+b{\sin }^{2}x}$

On differentiating and we get,

$\frac{dy}{dx}=\frac{(a{\cos }^{2}x+b{\sin }^{2}x)\frac{d}{dx}\sin 2x-\sin 2x\frac{d}{dx}(a{\cos }^{2}x+b{\sin }^{2}x)}{{(a{\cos }^{2}x+b{\sin }^{2}x)}^2}$

$=\frac{2\cos 2x(a{\cos }^{2}x+b{\sin }^{2}x)-\sin 2x(-a\sin 2x+b\sin 2x)}{{(a{\cos }^{2}x+b{\sin }^{2}x)}^2}$

$=\frac{2\cos 2x(a{\cos }^{2}x+b{\sin }^{2}x)-{\sin }^{2}2x(b-a)}{{(a{\cos }^{2}x+b{\sin }^{2}x)}^2}$

Hence, this is the answer.