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Math Help - Partial Derivatives

  1. #1
    Junior Member freestar's Avatar
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    Partial Derivatives

    Calculate x(\frac{\partial z}{\partial x})-y(\frac{\partial z}{\partial y}) for z=f(xy) where f is some differentiable function.

    I am not sure if I am doing it right. This is what I have tried so far.

    \frac{\partial z}{\partial x} = \frac{\partial f}{\partial x}(y)

    \frac{\partial z}{\partial y} = \frac{\partial f}{\partial y}(x)

    Help please..
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  2. #2
    A Plied Mathematician
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    Re: Partial Derivatives

    I find it always helpful to think of a function as a function of a certain number of independent variables. In this case, f is a function of one variable. I know it's got an x and a y in there, but if it were a function of two variables, I think you'd write as f(x,y) = g(xy), and even then, g would be a function of only one variable. How can that be? Because you could write a new variable u = xy, and then say that z = f(u).

    Hence, I think, rather, that

    \frac{\partial z}{\partial x}=y\,f'(xy), and so on. Does that make sense?
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  3. #3
    Junior Member freestar's Avatar
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    Re: Partial Derivatives

    Ahh makes sense. So the answer would this be the right answer?:
    xyf'(xy)-xyf'(xy) which would be equal to 0?
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  4. #4
    A Plied Mathematician
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    Re: Partial Derivatives

    Quote Originally Posted by freestar View Post
    Ahh makes sense. So the answer would this be the right answer?:
    xyf'(xy)-xyf'(xy) which would be equal to 0?
    Yep, that's what I get.
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