Question

Find $\frac{dy}{dx}$ of the function $(\cos x{)}^y=(\cos y{)}^x$ w.r.t.$x$.

Answer 1

$(\cos x{)}^y=(\cos y{)}^x$

apply log

$y\log \left(\cos x\right)=x\log \left(\cos y\right)=\frac{\log \left(\cos x\right)}{x}=\frac{\log \left(\cos y\right)}{y}$

$=\frac{dy}{dx}\log \left(\cos x\right)+\frac{y(-\sin x)}{\cos x}=\log \left(\cos y\right)+\frac{x(-\sin y)}{\cos y}.\frac{dy}{dx}$

$=\frac{dy}{dx}\left[\log \right(\cos x)+x\tan y]=\log \left(\cos y\right)+y\tan x$

$\Rightarrow \frac{dy}{dx}=\frac{\log \left(\cos y\right)+y\tan x}{\log \left(\cos x\right)+x\tan y}$