Question

Differentiate $\frac{x}{\sin x}$ with respect to $x$:

• A. $\frac{\sin x+x\cos x}{{\sin }^{2}x}$
• B. $\frac{\sin x-x\cos x}{{\sin }^{2}x}$
• C. $\frac{\sin x+x\cos x}{\sin x}$
• D. $\frac{\cos x-x\sin x}{\sin x}$

Answer 1

The correct option is B $\frac{\sin x-x\cos x}{{\sin }^{2}x}$

We need to differentiate $\frac{x}{\sin x}$

$\frac{d}{dx}\left(\frac{x}{\sin x}\right)=\frac{\sin x\frac{d}{dx}\left(x\right)-x\frac{d}{dx}\left(\sin x\right)}{{\left(\sin x\right)}^2}$
=$\frac{\left(\sin x\right)\left(1\right)-x\left(\cos x\right)}{{\sin }^{2}x}$

$=\frac{\sin x-x\cos x}{{\sin }^{2}x}$