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Math Help - derivative of exponential function

  1. #1
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    derivative of exponential function

    the problem is f(x)=x^(1/x).
    the answer is supposed to be something like x^(1/x-2)*(1-ln x)
    i attempted to solve it for like 20 mins and have no clue how to do this.
    can anyone help me plz?
    i want to know the steps of solving this.
    i have a midterm tmr and i'm definitely sure this type of problems will come up.

    thanks!!!!!!!
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  2. #2
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    Quote Originally Posted by katieeej View Post
    the problem is f(x)=x^(1/x).
    the answer is supposed to be something like x^(1/x-2)*(1-ln x)
    i attempted to solve it for like 20 mins and have no clue how to do this.
    can anyone help me plz?
    i want to know the steps of solving this.
    i have a midterm tmr and i'm definitely sure this type of problems will come up.

    thanks!!!!!!!
    1. Re-write the term of the function:

    x^{\frac1x} = e^{\frac1x \cdot \ln(x)}

    2. To calculate the derivation use chain rule (and product rule)

    3. f'(x)= e^{\frac1x \cdot \ln(x)} \cdot \left(\ln(x) \cdot (-x^{-2}) + \frac1x \cdot \frac1x  \right)

    4. Simplify this term.
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  3. #3
    Like a stone-audioslave ADARSH's Avatar
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    Quote Originally Posted by katieeej View Post
    the problem is f(x)=x^(1/x).
    the answer is supposed to be something like x^(1/x-2)*(1-ln x)
    i attempted to solve it for like 20 mins and have no clue how to do this.
    can anyone help me plz?
    i want to know the steps of solving this.
    i have a midterm tmr and i'm definitely sure this type of problems will come up.

    thanks!!!!!!!
    y = x^{1}/{x}



    ln(y) = {1}/{x} ln(x)


    Differentiate both sides

    {dy}/{y} = ( {-ln(x) }/{x^2} + {1}/{x^2} )dx


    Hence we put value of y after bringing on RHS

    {dy}/{dx} = x^(1/x) x^(-2) (1-ln x)



    EDIT too late
    Last edited by ADARSH; February 20th 2009 at 12:47 AM. Reason: Sometimes white looks more than usual(invisible)
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