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The value of $$\log _{e} 2 \frac{d}{d x}\left(\log _{\cos x} \operatorname{cosec} x\right)$$ at $$x=\frac{\pi}{4}$$ is

JEE · Math · previous-year question

  1. A.$$-2 \sqrt{2}$$
  2. B.$$2 \sqrt{2}$$
  3. C.$$-4$$
  4. D.4correct

Answer

D. 4

Explanation

Let $$f(x) = {\log _{\cos x}}\cos ec\,x$$ $$ = {{\log \cos ec\,x} \over {\log \cos x}}$$ $$ \Rightarrow f'(x) = {{\log \cos x\,.\,\sin x\,.\,\left( { - \cos ec\,x\cot x - \log \cos ec\,x\,.\,{1 \over {\cos x}}\,.\, - \sin x} \right)} \over {{{(\log \cos x)}^2}}}$$ at $$x = {\pi \over 4}$$ $$f'\left( {{\pi \over 4}} \right) = {{ - \log \left( {{1 \over {\sqrt 2 }}} \right) + \log \sqrt 2 } \over {{{\left( {\log {1 \over {\sqrt 2 }}} \right)}^2}}} = {2 \over {\log \sqrt 2 }}$$ $$\therefore$$ $${\log _e}2f'(x)$$ at $$x = {\pi \over 4} = 4$$

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