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Math Help - Partial Derivatives

  1. #1
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    Partial Derivatives

    Having a little trouble with this one:

    Find the first Partial Derivatives xy/x^2+y^2

    fx(x,y)= My interpretation was that in the numerator, x would be 1, leaving me with y

    and in the denominator x^2 would be 2x and y^2 would be zero, so I am left with

    y/2x, which is wrong. Not sure where I am mistaken so any help would be greatly appreciated. Thanks
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  2. #2
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    Re: Partial Derivatives

    You are looking at \displaystyle \frac{xy}{x^2+y^2} ?

    If so you have to use the quotient rule which is if \displaystyle y = \frac{u}{v} \implies y' = \frac{vu'-uv'}{v^2}

    Have you seen this rule before?
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  3. #3
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    Re: Partial Derivatives

    Yes, i am familiar with the quotient rule, but in this chapter they are only using "partial differentiation". The only example I have to go off of is f(x,y)= 3x - xy + 2xy f(x) (x,y)= 3-2xy + 4xy and f(y) (x,y) = -2xy + 2x So I am applying that same logic. Problem is I have no examples showing how to do a partial derivative of a quotient.
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  4. #4
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    Re: Partial Derivatives

    Ok, for \displaystyle f(x,y) = \frac{xy}{x^2+y^2} there are two first partial derivatives \displaystyle \frac{\partial f}{\partial x} and \displaystyle \frac{\partial f}{\partial y}

    Ill give the first one a go, you do the second.

    By the quotient rule \displaystyle \frac{\partial f}{\partial x} = \frac{(x^2+y^2)\times y- xy\times 2x}{(x^2+y^2)^2}

    How was that?
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  5. #5
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    Re: Partial Derivatives

    I was actually close, but not all the way there. Your example I had the same denominator, but in the numerator i had (x + y) * y - xy * (2x + 2y)

    f (y) (x,y)= x^3 -xy
    ----------
    (x + y)

    Thank you for you help it is greatly appreciated, and I do have a much better grasp of these problems now.
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  6. #6
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    Re: Partial Derivatives

    Looks like I need to learn to type a little better on here, sorry.
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  7. #7
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    Re: Partial Derivatives

    You can write equations using "tex" tags, hover your mouse over my equation to see the code.
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