Question

$y=\frac{2(x-\sin x{)}^{3/2}}{\sqrt{\left(x\right)}}$, find $\frac{dy}{dx}$

• A. $\frac{y}{2}(3.\frac{1-\cos x}{x-\sin x}-\frac{1}{x})$
• B. $\frac{y}{2}(2.\frac{1-\cos x}{x-\sin x}-\frac{1}{x})$
• C. $\frac{y}{2}(\frac{1-\cos x}{x-\sin x}-\frac{3}{x})$
• D. $\frac{y}{2}(\frac{1-\cos x}{x-\sin x}-\frac{2}{x})$

Answer 1

The correct option is A $\frac{y}{2}(3.\frac{1-\cos x}{x-\sin x}-\frac{1}{x})$

$y=\frac{2(x-\sin x{)}^{3/2}}{\sqrt{\left(x\right)}}$
Take log both sides,
$\Rightarrow \log y=\log 2+\frac{3}{2}\log (x-\sin x)-\frac{1}{2}\log x$
Differentiating both sides w.r.t $x$
$\Rightarrow \frac{1}{y}\frac{dy}{dx}=\frac{3}{2}\frac{1-\cos x}{x-\sin x}-\frac{1}{2x}$
$\Rightarrow \frac{dy}{dx}=\frac{y}{2}(3.\frac{1-\cos x}{x-\sin x}-\frac{1}{x})$