Question

If $y={\left(\sin x\right)}^{{\left(\sin x\right)}^{\left(\sin x\right)....\infty }},$ then $\frac{dy}{dx}=$

• A. $\frac{y^2}{\sin x\left(1-\log y\right)}$
• B. $\frac{y^2\sin x}{1-\log y}$
• C. $\frac{y^2\cot x}{1-\log y}$
• D. $\frac{y^2\tan x}{1-\log y}$

Answer 1

Answer: (C)

$y={\left(\sin x\right)}^y\Rightarrow \log y=y\log \sin x$
$\Rightarrow \frac{1}{y}\frac{dy}{dx}=\frac{dy}{dx}\log \left(\sin x\right)+y\frac{1}{\sin x}.\cos x$
$\Rightarrow \frac{dy}{dx}\left(\frac{1}{y}-\log \sin x\right)=y\cot x$
$\Rightarrow \frac{dy}{dx}=\frac{y^2\cot x}{1-y\log \sin x}=\frac{y^2\cot x}{1-\log y}\left[\because \log y=\log \sin x\right]$