Question

Differentiate w.r.to $x$

$\frac{3\cos x-4}{{\sin }^{2}x}$

Answer 1

$y=\frac{3\cot x-4}{{\sin }^{2}x}$

$y=\frac{f\left(x\right)}{g\left(x\right)}$ then

$(\because \frac{dy}{dx}=\frac{f^{\prime }\left(x\right)g\left(x\right)-g^{\prime }\left(x\right)f\left(x\right)}{g(x{)}^2})$

$\therefore \frac{dy}{dx}=\frac{-3\sin x\left({\sin }^{2}x\right)-\left(2\sin x\cos x\right)(3\cos x-4)}{{\sin }^{4}x}$

$=\frac{-3\sin x\left({\sin }^{2}x\right)-\left(6{\cos }^2\right)+\left(8\cos x\right)}{{\sin }^{3}x}$

$\therefore \frac{dy}{dx}=\frac{-3si{n}^{3}x-\left(6{\cos }^2\right)+\left(8\cos x\right)}{{\sin }^{3}x}$