Question

If ${\left(\cos x\right)}^y={\left(\cos y\right)}^x,\mathrm{find}\frac{dy}{dx}$

Answer 1

$$\begin{gathered} {\left(\cos x\right)}^y={\left(\cos y\right)}^x \\ \mathrm{Taking}\log \mathrm{on}\mathrm{both}\mathrm{sides}\mathrm{we}\mathrm{get} \\ y\log \cos x=x\log \cos y \\ \Rightarrow \frac{dy}{dx}\log \cos x-y\tan x=\log \cos y- \\ \Rightarrow \frac{dy}{dx}\mathrm{logcos}x+x\tan y\frac{dy}{dx}=\mathrm{logcos}y+y\tan x \\ \Rightarrow \frac{dy}{dx}\left(\log \cos x+x\tan y\right)=\mathrm{logcos}y+y\tan x \\ \Rightarrow \frac{dy}{dx}=\frac{\mathrm{logcos}y+y\tan x}{\mathrm{logcos}x+x\tan y} \end{gathered}$$