x^y=y^x find an expression for dy/dx in terms of y and x

June Mejia

June Mejia

Open question

2022-08-16

x y = y x
find an expression for d y / d x in terms of y and x
for the first part is y x y 1 and for the second part i got ln ( y ) y x d y / d x
when solving for dy/dx i got y x y 1 ) / ( ln ( y ) y x )
Just was wondering if my work is correct?
Sorry for the bad formatting, Im not sure how to make it formatted nice.

Answer & Explanation

Donovan Shields

Donovan Shields

Beginner2022-08-17Added 13 answers

It might be easier to take log first and work with the expression. We have
y log x = x log y
Now differentiating we get
y log x + y x = log y + x y y y ( log x x y ) = ( log y y x )
Hence, we get that
y = ( log y y x ) ( log x x y )

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