Question

If $y={\log }_{\sin x}\left(\tan x\right),$ then $\left(dy/dx\right)\pi /4=$

• A. $4/\log 2$
• B. $-4/\log 2$
• C. $-4/(\log 2)$
• D. None of these.

Answer 1

Answer: (A)

$y=\frac{\log \tan x}{\log \sin x}\Rightarrow \frac{dy}{dx}$
$=\frac{\log \left(\tan x\right).\frac{1}{\sin x}.\cos x-\log \left(\sin x\right).\frac{1}{\tan x}.{\sin }^{2}x}{{\left(\log \sin x\right)}^2}$
At $x=\frac{x}{4}$
$\frac{dy}{dx}=\log \left(1\right).1-\log \left(\frac{\frac{1}{\sqrt{2}}.2}{{\left(\log \left(\frac{1}{\sqrt{2}}\right)\right)}^2}\right)$
$=\frac{\frac{2}{2}\log 2}{\frac{1}{4}{\left(\log 2\right)}^2}=\frac{4}{\log 2}$