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. $x^{\sin\,x}\left(\cos\,x\, \log\,x+\frac{\sin\,x}{x}\right)$
• D. $\cos\,x\, \log\,x+\frac{\sin\,x}{x}$

Answer 1

The Correct Option is C

The correct option is(C): \(x^{\sin x}\left(\frac{\sin x}{x}+\log x \cos x\right)\).

Given, \(y=x^{\sin x}\)
On taking log both sides, we get
\(\log y=\sin x \log x\)
On differentiating both sides w.r.t. \(x\), we get 
\(\frac{1}{y} \cdot \frac{d y}{d x}= \sin x \frac{d}{d x} \log x+\log x \frac{d}{d x} \sin x\)
\(\Rightarrow \frac{1}{y} \cdot \frac{d y}{d x}=\sin x \cdot \frac{1}{x}+\log x \cdot \cos x\)
\(\Rightarrow \frac{d y}{d x}=y\left(\frac{\sin x}{x}+\log x \cdot \cos x\right)\)
\(\Rightarrow \frac{d y}{d x}=x^{\sin x}\left(\frac{\sin x}{x}+\log x \cos x\right)\)