Question

Find `(dy)/(dx)`, if (cos x)^y = (cos y)^x. 

Answer 1

Given, (cos x)^y = (cos y)^x

Taking log on both sides, we get 

y log(cos x) = x log(cos y)

Differentiating both sides w.r.t. x, we get

`(dy)/(dx).log(cosx) + y.((-sin x)/(cos x)) = 1.log(cos y) + x.((-sin y)/(cos y)) (dy)/(dx)`

`(dy)/(dx).log(cosx) - (y sin x)/(cos x) = log (cos y) - (x sin y)/(cos y) "dy"/"dx"`

`(dy)/(dx)[log (cos x) + (y sin y)/(cos y)] = log (cos y) + (y sinx)/(cos x)`

`(dy)/(dx)(log cos x + x tan y) = log (cos y) + y tan x`

`(dy)/(dx) = (log (cos y) + y tan x)/(log (cos x) + tan y)`