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Question
If \[y = \left( \cos x \right)^{\left( \cos x \right)^{\left( \cos x \right) . . . \infty}}\],prove that \[\frac{dy}{dx} = - \frac{y^2 \tan x}{\left( 1 - y \log \cos x \right)}\]?
Solution
\[\text{ We have, y} = \left( \cos x \right)^{\left( \cos x \right)^{\left( \cos x \right)^{. . . . \infty}}} \]
\[ \Rightarrow \log y = y \log\left( \cos x \right)\]
\[\frac{1}{y}\frac{dy}{dx} = y\frac{d}{dx}\left\{ \log \cos x \right\} + \log \cos x\frac{dy}{dx}\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = y\left( \frac{1}{\cos x} \right)\frac{d}{dx}\left( \cos x \right) + \log \cos x\frac{dy}{dx}\]
\[ \Rightarrow \frac{dy}{dx}\left( \frac{1}{y} - \log \cos x \right) = \frac{y}{\cos x}\left( - \sin x \right)\]
\[ \Rightarrow \frac{dy}{dx}\left( \frac{1 - y \log \cos x}{y} \right) = - y \tan x\]
\[ \Rightarrow \frac{dy}{dx} = - \frac{y^2 \tan x}{\left( 1 - y \log \cos x \right)}\]
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