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Math Help - Logarithmic Differentiation

  1. #1
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    Logarithmic Differentiation

    Hi, I'm having a bit of trouble with my homework in some parts:

    1)

    Find y' if:

    x^y = y^x

    I have no idea on how to even start this.

    As for the other:

    2)

    Use logarithmic differentiation:

    sqrt(x)*(e^(x^2))*((x^2)+1)^10

    My work:

    log both sides:

    log y = log(sqrt(x)*(e^(x^2))*((x^2)+1)^10

    1/y * dy/dx = 1/2 * ln x + ln e^(x^2) + 10ln((x^2)+1)

    Then I am confused..

    Thank-you very much!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Prim3 View Post
    Hi, I'm having a bit of trouble with my homework in some parts:

    1)

    Find y' if:

    x^y = y^x

    I have no idea on how to even start this.
    Take logs:

    y \ln(x)=x \ln(y)

    Now differentiate with respect to x :

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

    Now simplify and rearrange:

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

    CB
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