Question

Derivative of $f\left(x\right)=\frac{\sin x-x\cos x}{(x\sin x+\cos x)}$ w.r.t. $x$ is

• A. $\frac{x\sin x}{(x\sin x+\cos x{)}^3}$
• B. $\frac{x}{(x\sin x+\cos x{)}^3}$
• C. $\frac{x^2}{(x\sin x+\cos x{)}^2}$
• D. $\frac{x^2\sin x}{(x\sin x+\cos x{)}^2}$

Answer 1

The correct option is C $\frac{x^2}{(x\sin x+\cos x{)}^2}$

$f\left(x\right)=\frac{\sin x-x\cos x}{(x\sin x+\cos x)}\Rightarrow f^{\prime }\left(x\right)=\frac{(x\sin x+\cos x)(\cos x-\cos x+x\sin x)-(x\cos x+\sin x-\sin x)(\sin x-x\cos x)}{(x\sin x+\cos x{)}^2}\Rightarrow f^{\prime }\left(x\right)=\frac{(x\sin x+\cos x)\left(x\sin x\right)-\left(x\cos x\right)(\sin x-x\cos x)}{(x\sin x+\cos x{)}^2}\Rightarrow f^{\prime }\left(x\right)=\frac{x^2{\sin }^{2}x+x\cos x\sin x-x\cos x\sin x+x^2{\cos }^{2}x}{(x\sin x+\cos x{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{x^2}{(x\sin x+\cos x{)}^2}$