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If y^(x) = x^(y) , "then" (dy)/(dx) is ...

If ` y^(x) = x^(y) , "then" (dy)/(dx)` is equal to

A

`(y)/(x) ((y + x logy)/(x - y logx))`

B

`(y)/(x) ((y - x logy)/(x - y logx))`

C

`(y)/(x) ((y + x logy)/(x + y logx))`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
B
Given , `y^(x) = x^(y)`
On taking log on both sides , we get
` log y^(x) = log x^(y)`
`rArr x log y = y log x`
On differentiating both sides w.r.t.x, we get
`x((1)/(y)) (dy)/(dx)+(logx) = y(1)/(x) + (logx)(dy)/(dx)`
`rArr (x)/(y)(dy)/(dx)- (log x) = (y)/(x) - logy`
`rArr ((x)/(y) - log x) (dy)/(dx) = (y)/(x) - log y`
`rArr ((x - y log x )/(y) (dy)/(dx) = (y -x log y)/(x)`
` rArr (dy)/(dx) = (y)/(x) ((y-x log y)/(x -y log x ))`
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