The value of $\log _{e} 2 \frac{d}{d x}\left(\log _{\cos x} \operatorname{cosec} x\right)$ at $x=\frac{\pi}{4}$ is
Solution
<p>Let $f(x) = {\log _{\cos x}}\cos ec\,x$</p>
<p>$= {{\log \cos ec\,x} \over {\log \cos x}}$</p>
<p>$$ \Rightarrow f'(x) = {{\log \cos x\,.\,\sin x\,.\,\left( { - \cos ec\,x\cot x - \log \cos ec\,x\,.\,{1 \over {\cos x}}\,.\, - \sin x} \right)} \over {{{(\log \cos x)}^2}}}$$</p>
<p>at $x = {\pi \over 4}$</p>
<p>$$f'\left( {{\pi \over 4}} \right) = {{ - \log \left( {{1 \over {\sqrt 2 }}} \right) + \log \sqrt 2 } \over {{{\left( {\log {1 \over {\sqrt 2 }}} \right)}^2}}} = {2 \over {\log \sqrt 2 }}$$</p>
<p>$\therefore$ ${\log _e}2f'(x)$ at $x = {\pi \over 4} = 4$</p>
About this question
Subject: Mathematics · Chapter: Differentiation · Topic: Derivatives of Standard Functions
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