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Math Help - Partial Derivative - Chain Rule

  1. #1
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    Partial Derivative - Chain Rule

    I have this problem which I am stuck on. It tells me to find the derivative of the function in 2 ways...first by substitution then by the chain rule using partial derivatives. I found the first one fine using substitution then taking a derivative. The second one I came up with a different answer (which I believe is incorrect). Can someone help me get unstuck please? I've attached a picture of my work. Thanks
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Partial Derivative - Chain Rule

    I didn't check whole the answer but I notice a mistake by calculating \frac{dz}{dy} with the chain rule:
    \frac{d}{dy}\left(\frac{x}{x^2+y^2}\right)=x\cdot \frac{d}{dy}(x^2+y^2)^{-1}
    =-x\cdot(x^2+y^2)^{-2}\cdot 2y =\frac{-2xy}{(x^2+y^2)^2}

    Which is different from your answer.
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  3. #3
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    Re: Partial Derivative - Chain Rule

    That was it. Dumb error. Thanks for the help!
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    MHF Contributor Siron's Avatar
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    Re: Partial Derivative - Chain Rule

    You're welcome!
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