Question

$\frac{d\left\{{\left(\sin \left(x\right)\right)}^{\log \left(x\right)}\right\}}{dx}=$

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

Answer 1

The correct option is B

${\left(\sin \left(x\right)\right)}^{\log \left(x\right)}\left\{\left[\frac{\log \left(\sin \left(x\right)\right)}{x}\right]+\cot \left(x\right)\log \left(x\right)\right\}$

Explanation for correct option:

Step-1: Simplify the given data.

Given, $\frac{d\left\{{\left(\sin \left(x\right)\right)}^{\log \left(x\right)}\right\}}{dx}$

Let, $y={\left(\sin \left(x\right)\right)}^{\log \left(x\right)}$

Taking log both side.

$$\begin{gathered} \Rightarrow \log \left(y\right)=\log \left(\sin {\left(x\right)}^{\log \left(x\right)}\right) \\ \Rightarrow \log \left(y\right)=\log \left(x\right)\times \log \left(\sin \left(x\right)\right)\left(\because \log {\left(a\right)}^x=x\log \left(a\right)\right) \end{gathered}$$

Step-2: Differentiate w.r.t $x$

$\Rightarrow \frac{1}{y}\times \frac{dy}{dx}=\frac{1}{x}\times \log \left(\sin \left(x\right)\right)+\log \left(x\right)\frac{1}{\sin \left(x\right)}\times \cos \left(x\right)$

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=y\left[\frac{\log \left(\sin \left(x\right)\right)}{x}+\log \left(x\right)\cot \left(x\right)\right] \\ \Rightarrow \frac{dy}{dx}={\left(\sin \left(x\right)\right)}^{\log \left(x\right)}\left[\frac{\log \left(\sin \left(x\right)\right)}{x}+\log \left(x\right)\cot \left(x\right)\right] \end{gathered}$$

Hence, the correct answer is option is $\left(B\right)$