Question

Find `("d"y)/("d"x)`, if x^y = y^x 

Answer 1

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"/("d"x)(y log x) = "d"/("d"x)(x log y)`

∴ `y*"d"/("d"x)(log x) + "d"/("d"x)(y) = x*"d"/("d"x)(log y) + log y* "d"/("d"x)(x)`

∴ `y*1/x + logx*("d"y)/("d"x) = x*1/y*("d"y)/("d"x) + logy*1`

∴ `logx ("d"y)/("d"x) - x/y*("d"y)/("d"x) = logy - y/x`

∴ `((ylogx - x)/y) ("d"y)/("d"x) = (xlogy - y)/x`

∴ `("d"y)/("d"x) = (xlogy - y)/x xx y/(ylogx - x)`

∴ `("d"y)/("d"x) = y/x((xlogy - y)/(ylogx - x))`