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

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

A

`y x^(y-1)`

B

`(y^(2))/(x(1-y log x))`

C

` y/(x(1+y log x))`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
B
Given
` y = x^(y)`
Taking log on both sides, we get
log y = y log x
` Rightarrow 1/y .(dy)/(dx) =y .1/x + (dy)/(dx) . Log x`
` Rightarrow (dy)/(dx) [ 1/y - log x] = y/x `
` Rightarrow (dy)/(dx) = (y^(2))/(x(1-y log x))`
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Knowledge Check

  • Let y = x^(x^(x...oo))," then " (dy)/(dx) is equal to

    A
    `yx^(y-1)`
    B
    `(y^(2))/(x(1-y log x))`
    C
    `(y)/(x (1 + y log x ))`
    D
    `None of these

  • If y=x^(n)logx+x(logx)^(n)," then "(dy)/(dx) is equal to

    A
    `x^(n-1)(1+nlogx)+(logx)^(n+1)[n+logx]`
    B
    `x^(n-2)(1+nlogx)+(logx)^(n-1)[n+logx]`
    C
    `n^(n-1)(1+nlogx)+(logx)^(n-1)[n-logx]`
    D
    None of the above

  • 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

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