Question

The derivative of $y=x^{\sin x}$ is

• A. $x^{\sin x}(\cos x\log x+\frac{\sin x}{x})$
• B. $\cos x\log x+\frac{\sin x}{x}$
• C. $\cos x{x}^{\sin x-1}$
• D. $\frac{\sin 2x}{2}{x}^{\sin x-1}$

Answer 1

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

Given, $y=x^{\sin x}$
On taking $log$ both sides, we get
$\log y=\sin x\log x$
On differentiating both sides with respect to $x$, we get
$\frac{1}{y}\cdot \frac{dy}{dx}=\sin x\frac{d}{dx}\log x+\log x\frac{d}{dx}\sin x$
$\Rightarrow \frac{1}{y}\cdot \frac{dy}{dx}=\sin x\cdot \frac{1}{x}+\log x\cdot \cos x$
$\Rightarrow \frac{dy}{dx}=y(\frac{\sin x}{x}+\log x\cdot \cos x)$
$\Rightarrow \frac{dy}{dx}=x^{\sin x}(\frac{\sin x}{x}+\log x\cos x)$.