Question

Derivative of ${\left(x\cos x\right)}^x$ with respect to $x$ is

• A. ${\left(x\cos x\right)}^x\left[\right(\log x+1)-\{\log \cos x+\frac{x}{\cos x}.(\sin x\left)\right\}]$
• B. ${\left(x\cos x\right)}^x\left[\right(\log x+1)+\{\log \cos x+\frac{x}{\cos x}.(-\sin x\left)\right\}]$
• C. ${\left(x\cos x\right)}^x\left[\right(\log x+1)+\{\log \sin x+\frac{x}{\cos x}.(\cos x\left)\right\}]$
• D. None of these

Answer 1

Answer: (B)

${\left(x\cos x\right)}^x\left[\right(\log x+1)+\{\log \cos x+\frac{x}{\cos x}.(-\sin x\left)\right\}]$

Let $y={\left(x\cos x\right)}^x$.
Taking logarithm on both sides, we get
$\log y=\log {\left(x\cos x\right)}^x$
$=x\log \left(x\cos x\right)$
$=x\log x+x\log \cos x$
Differentiating both sides with respect to $x$, we get
$\frac{1}{y}\frac{dy}{dx}=\frac{d}{dx}\left(x\log x\right)+\frac{d}{dx}\left(x\log \cos x\right)$
$\frac{dy}{dx}=y[(1+\log x)+\left(\log \cos x+\frac{x}{\cos x}\frac{d}{dx}\left(\cos x\right)\right)]$
$\therefore \frac{dy}{dx}={\left(x\cos x\right)}^x\left[\right(\log x+1)+\{\log \cos x+\frac{x}{\cos x}.(-\sin x\left)\right\}]$