Results 1 to 2 of 2

Math Help - x^y = y^x

  1. #1
    Junior Member
    Joined
    Jan 2009
    Posts
    57

    x^y = y^x

    What is the derivative of y when x^y = y^x?

    Can someone please give me the steps to solving this equation?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Oct 2009
    Posts
    40
    The first step is to take the natural log of both sides:

    ln(x^y) = ln(y^x)        \therefore      yln(x) = xln(y)

    Taking the derivative of both sides with respect to x, using the product and chain rules, gets:

    y \frac{d}{dx}ln(x) + ln(x){dy}{dx}= x \frac {d}{dx}ln(y) + ln(y){d}{dx}x

    \frac{y}{x} + ln(x) \frac{dy}{dx}= x \frac {d}{dy}ln(y) \frac{dy}{dx} + ln(y)

    \frac{y}{x} + ln(x) \frac{dy}{dx} = \frac{x}{y} \frac {dy}{dx} + ln(y)

    \frac{dy}{dx} = \frac{ln(y) - \frac{y}{x}}{ln(x) - \frac{x}{y}}
    Follow Math Help Forum on Facebook and Google+


/mathhelpforum @mathhelpforum