Differentiate without using chain rule in 5 steps

I'm facing problem with this challenge problem. Evaluate

$\frac{d}{dx}\frac{\left(x\sin x+\cos x\right)}{\left(x\sin x-\cos x\right)^2}$ without using chain rule and using the basic differentiation rules in 5 steps.

• What problems have you faced while trying this? What basic differentiation rules do you know? – B. Pasternak Nov 18 '17 at 9:00
• It appears to be a lot hairy and difficult to simply using product and quotient rules. – Paras Khosla Nov 18 '17 at 9:06
• It's really not so bad if you just do it slowly and properly. The point is that you have to show where you get stuck, not ask others to do it for you. – B. Pasternak Nov 18 '17 at 9:18
• That is what I'm not able to make out, I've expanded all the terms but at the end I reach a point where no further simplification is possible and I don't get the desired answer. – Paras Khosla Nov 18 '17 at 9:22
• Write it down here, then perhaps somebody can help. – B. Pasternak Nov 18 '17 at 9:51

the quotient rule is given by $$\frac{u'v-uv'}{v^2}$$ given $$u=x\sin(x)+\cos(x)$$ $$v=?(x\sin(x)-\cos(x))^2$$ then we get $$u'=\sin(x)+x\cos(x)-\sin(x)=x\cos(x)$$ $$v'=2(x\sin(x)-\cos(x))(\sin(x)+x\cos(x)+\sin(x))$$ can you proceed?