Question

Find `(dy)/(dx)`, if x^y = y^x 

Answer 1

Given x^y = y^x 

Taking logarithm of both sides, we get

log x^y = log y^x

∴ y log x = x log y

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

`d/(dx)(ylogx) = d/(dx)(xlogy)`

∴ `y.d/(dx)(logx) + d/(dx)(y) = x.d/(dx)(logy) + logy. d/(dx)(x)`

∴ `y. 1/x + logx.(dy)/(dx) = x. 1/y.(dy)/(dx) + logy.1`

∴ `(logx - x/y)(dy)/(dx) = (logy - y/x)`

∴ `((ylogx - x)/y) (dy)/(dx) = (xlogy - y)/x`

∴ `(dy)/(dx) = ((xlogy - y)/x) xx (y/(ylogx - x))`

∴ `(dy)/(dx) = y/x((xlogy - y)/(ylogx - x))`