Question

Differentiate

$\frac{3\sin 2x}{5+4\cos 2x}$

Answer 1

Let $y=\frac{3\sin 2x}{5+4\cos 2x}$

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

$\therefore \frac{dy}{dx}=\frac{6\cos 2x(5+4\cos 2x)-(-8\sin 2x)\left(3\sin 2x\right)}{(5+4\cos 2x{)}^2}$

$=\frac{30\cos 2x+24{\cos }^{2}2x+24{\sin }^{2}2x}{(5+4\cos 2x{)}^2}$

$\frac{dy}{dx}=\frac{24+30\cos 2x}{(5+4\cos 2x{)}^2}$

$\therefore \frac{dy}{dx}=\frac{6(4+5\cos 2x)}{(5+4\cos 2x{)}^2}$