first derivative from x^y = y^x

Find the first derivative from

x^y = y^x

Answare:

Use logarithmic derivative to find the first derivative this function

ln x^y = ln y^x

y ln x = x ln y

y = (x ln⁡y) / ln⁡x

so:

d/dx (x ln⁡ y) = d/dx (x) ln y + x d/dx (ln y)

= ln y + x [d/dy (ln y)  dy/dx]

= ln y + x ( 1/y) dy/dx

 = ln y +  x/y dy/dx

So:

y = (x ln⁡y) / ln ⁡x

dy/dx = [ d/dx(x lny) ln⁡ x – x ln⁡ y d/dx (ln⁡ x) ] / (ln⁡ x)^2

dy/dx = [ ( ln y + x/y dy/dx) ln⁡ x – x ln⁡ y 1/x] / (ln⁡ x)^2

dy/dx = [ ln⁡ y ln⁡ x + ln⁡x ( x/y)  dy/dx –  ln⁡ y ] / (ln⁡ x)^2

ln^2 x dy/dx =  ln⁡ y ln⁡ x + ln⁡x ( x/y) dy/dx –  ln⁡ y

ln^2 x dy/dx – x/y ln⁡ x dy/dx = ln⁡ y ln⁡ x – ln ⁡y

dy/dx (ln^2 x  –  x/y ln⁡x ) = ln⁡ y (ln⁡x – 1)

dy/dx = [ln⁡ y ln⁡ (x-1)] / [ln^2 x – x/y ln⁡ x]

 

 

 

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Find the first derivative from x^y = y^x