Question

Derivative of ${\log x}^{\cos x}$ with respect to $x$ is

• A. ${\log x}^{\cos x}[\frac{\cos x}{x\log x}-\sin x\log \left(\log x\right)]$
• B. ${\log x}^{\cos x}[\frac{\cos x}{x\log x}-\cos x\log \left(\log x\right)]$
• C. ${\log x}^{\sin x}[\frac{\sin x}{x\log x}-\sin x\log \left(\log x\right)]$
• D. None of these

Answer 1

Answer: (D)

None of these

Let $y=\log x^{\cos x}$

$\Rightarrow y=\left(\cos x\right)\left(\log x\right)$

$\therefore \frac{dy}{dx}=\left(\cos x\right)\left(\frac{1}{x}\right)+(-\sin x)\left(\log x\right)$

$=\frac{\cos x}{x}-\sin x\log x$