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If x ^(y) = e ^( x -y) , then show that...

If `x ^(y) = e ^( x -y) ,` then show that ` (dy)/(dx) = (log x )/( (1 + log x ) ^(2))`

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If x=tan(e^(-y)) , then show that (dy)/(dx)=(-e^y)/(1+x^2) .

If x ^(y) + y ^(x) = a ^(b) then show that (dy)/(dx) =- ((y x ^(y-1) + y^(x) log y)/( x ^(y) log x + x y ^(x -1)))

Knowledge Check

  • If y = log _(cos x ) sin x, then , (dy)/(dx ) =

    A
    `(cot x log cos x + tan x log sin x) //(log cos x )^(2)`
    B
    `(tan x log cos x + cot x log sin x) //(log cos x )^(2)`
    C
    `cot x log cos x + tan x log sin x) // (log sin x) ^(2)`
    D
    none

  • A: If y = x ^(y) then (dy)/(dx) = (y ^(2))/(x(1- log y )) If y = f (x) ^(y), then (dy)/(dx) = (y ^(2) f '(x))/(f (x) [1- ylog f (x)])= (y ^(2) f'(x))/(f (x) [1- log y])

    A
    Both A and R are true R is correct reason of A
    B
    Both A and R are treu R is not correct reason of A
    C
    A is true but R is false
    D
    A is false but R is true

  • The solution of (dy)/(dx) = (x log x^(2) + x)/(sin y + y cos y) is

    A
    `y sin y = x^(2) log x + c`
    B
    `y sin y = x^(2) + c`
    C
    `y sin y = x^(2) + log x + c`
    D
    `y sin y = x log x + c`

    • Similar Questions

    Explore conceptually related problems

    If x ^( log y) = log x, then prove that (dy)/(dx) = (y)/(x) ((1- log x log y)/( (log x) ^(2)))

    x (dy)/(dx) + 2y = x^(2)log x

    x log x(dy)/(dx) + y = (2)/(x)log x

    Find the derivatives of the function If x = log (1 + sqrty), then show that (dy)/(dx) = 2 sqrtye ^(x)

    If x^2+y^2=27xy , then show that log((x-y)/(5))=1/2[logx+logy] .

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