Question

If  \[\left( \cos x \right)^y = \left( \cos y \right)^x , \text{ find } \frac{dy}{dx}\] ?

 

Answer 1

\[\left( \cos x \right)^y = \left( \cos y \right)^x \]
\[\text{ Taking log on both sides we get }\]
\[y\log\cos x = x\log\cos y \]
\[ \Rightarrow \frac{dy}{dx}\log\cos x - y\tan x = \log\cos y -  x \tan y\]
\[ \Rightarrow \frac{dy}{dx}\log\ cos x + x\tan y\frac{dy}{dx} = \log\cos y + y\tan x \]
\[ \Rightarrow \frac{dy}{dx}\left( \log\cos x + x\tan y \right) = log\cos y + y\tan x\]
\[ \Rightarrow \frac{dy}{dx} = \frac{\log\cos y + y\tan x}{\log \cos x + x\tan y}\]