Question

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

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

Answer 1

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

$y=x^{\sin x}\Rightarrow \ln y=\sin x\ln x$
Differentiating both side
$\frac{1}{y}\frac{dy}{dx}=\frac{1}{x}\sin x+\cos x\ln x$
$\Rightarrow \frac{dy}{dx}=x^{\sin x}(\cos x\log x+\frac{\sin x}{x})$