Question

The value of $\log_e2\frac{d}{dx}(\log_{cos⁡ x}\cosec x)$ at $x=\frac{\pi}{4}$ is

• A. $-2\sqrt2$
• B. $2\sqrt2$
• C. $-4$
• D. $4$

Answer 1

Answer: (D)

$4$

Answer 2

The correct answer is (D) : 4
Let
$f(x) = \log_{cos x}\cosec x$
$= \frac{\log \cosec x/}{\log \cos x}$
$f'(x) = \frac{\log \cos x.\sin x.(-\cosec x \cot x-\log \cosec x . \frac{1}{\cos x}-\sin x)}{(\log\cos x)^2}$
at x $= \frac{π}{4}$
$f'(\frac{\pi}{4}) = \frac{-log(\frac{1}{\sqrt{2}})+log\sqrt2}{(log\frac{1}{\sqrt2})^2}$
$= \frac{2}{log\sqrt{2}}$
$∴ \log_e 2f' (x)\ at\ x = \frac{\pi}{4} = 4$