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

The Correct Option is 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] $