Question

If $y={\log }_{\cos x}\left(\tan x\right)$, then $\frac{dy}{dx}{|}_{x=\frac{\pi }{4}}$ is equal to

• A. $\frac{-2}{\ln \sqrt{2}}$
• B. $\frac{2}{\ln \sqrt{2}}$
• C. $\frac{-\sqrt{2}}{\ln \sqrt{2}}$
• D. $\frac{\sqrt{2}}{\ln \sqrt{2}}$

Answer 1

The correct option is A $\frac{-2}{\ln \sqrt{2}}$

$x=\frac{\pi }{4}\Rightarrow y=0$
We can write,
$y={\log }_{\cos x}\left(\tan x\right)\Rightarrow y={\log }_{\cos x}\frac{\sin x}{\cos x}\Rightarrow y={\log }_{\cos x}\sin x-1\Rightarrow y+1=\frac{\ln \sin x}{\ln \cos x}\Rightarrow (y+1)\left(\ln \cos x\right)=\ln \sin x$
Differentiating w.r.t. $x$,
$\Rightarrow (y+1)\times \frac{-\sin x}{\cos x}+y^{\prime }\left(\ln \cos x\right)=\frac{\cos x}{\sin x}$
Putting $x=\frac{\pi }{4},y=0$
$\Rightarrow -1-y^{\prime }\left(\ln \sqrt{2}\right)=-1\Rightarrow y^{\prime }=-\frac{2}{\ln \sqrt{2}}$