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

The Correct Option is D

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 \(= \frac{π}{4}\)
\(f'(\frac{\pi}{4}) = \frac{-log(\frac{1}{\sqrt{2}})+log\sqrt2}{(log\frac{1}{\sqrt2})^2}\)
\(= \frac{2}{log\sqrt{2}}\)
\(∴ \log_e 2f' (x)\ at\ x = \frac{\pi}{4} = 4\)