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} $
$\frac{dy}{dx} =\log\left(1\right) .1 -\log \left(\frac{\frac{1}{\sqrt{2}}. 2}{\left(\log \left(\frac{1}{\sqrt{2}}\right)\right)^{2}}\right)$
$= \frac{\frac{2}{2} \log2}{\frac{1}{4} \left(\log2\right)^{2}} = \frac{4}{\log2}$