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Math Help - Chain Rule in Partial Derivative

  1. #1
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    Chain Rule in Partial Derivative

    f\left(u,v,w\right) is differentiable and u=\;x-y,\; v=\;y-z,\;w=\;z-x, show that \frac{\partial{f}}{\partial{x}}\;+\;\frac{\partial  {f}}{\partial{y}}\;+\;\frac{\partial{f}}{\partial{  z}}\;=\;0.

    I think i'll use chain rule and that equation will be \frac{\partial{f}}{\partial{u}}\frac{\partial{u}}{  \partial{x}}\;+\;\frac{\partial{f}}{\partial{v}}\f  rac{\partial{v}}{\partial{y}}\;+\;\frac{\partial{f  }}{\partial{w}}\frac{\partial{w}}{\partial{z}}\;=\  ;0
    From now on i can not find any value for \frac{\partial{f}}{\partial{u}}\; ,, \frac{\partial{f}}{\partial{v}}\; and \;\frac{\partial{f}}{\partial{x}}. So how can i find these values?

    If that's not the way of solution, i need some hints.
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  2. #2
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    This is only intuition

    f\left(u,v,w\right) = f\left(x-y,y-z,z-x\right)

    \frac{\partial f}{\partial u} = \frac{\partial f}{\partial (x - y)} = \frac{\partial f}{\partial x} - \frac{\partial f}{\partial y}.
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  3. #3
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    wow.! I didn't know, i can split derivative like that.
    Thanks. =)
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  4. #4
    Member Ruun's Avatar
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    Neither do I lol, it`s just the only thing I can think of, may be wrong. Wait for a serious mathematician or something..
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  5. #5
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    Lightbulb Hint

    Hint: \frac{{\partial f}}{{\partial x}} = \frac{{\partial f}}{{\partial u}} \times \frac{{\partial u}}{{\partial x}} + \frac{{\partial f}}{{\partial w}} \times \frac{{\partial w}}{{\partial x}}<br />
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  6. #6
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    Quote Originally Posted by Lafexlos View Post
    I think i'll use chain rule and that equation will be \frac{\partial{f}}{\partial{u}}\frac{\partial{u}}{  \partial{x}}\;+\;\frac{\partial{f}}{\partial{v}}\f  rac{\partial{v}}{\partial{y}}\;+\;\frac{\partial{f  }}{\partial{w}}\frac{\partial{w}}{\partial{z}}\;=\  ;0
    From now on i can not find any value for \frac{\partial{f}}{\partial{u}}\; ,, \frac{\partial{f}}{\partial{v}}\; and \;\frac{\partial{f}}{\partial{x}}. So how can i find these values?

    If that's not the way of solution, i need some hints.
    .
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  7. #7
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    \frac{{\partial f}}{{\partial x}} = \frac{{\partial f}}{{\partial u}} \times \frac{{\partial u}}{{\partial x}} + \frac{{\partial f}}{{\partial w}} \times \frac{{\partial w}}{{\partial x}} = \frac{{\partial f}}{{\partial u}} - \frac{{\partial f}}{{\partial w}}

    \frac{{\partial f}}{{\partial y}} = \frac{{\partial f}}{{\partial v}} \times \frac{{\partial v}}{{\partial y}} + \frac{{\partial f}}{{\partial u}} \times \frac{{\partial u}}{{\partial y}} = \frac{{\partial f}}{{\partial v}} - \frac{{\partial f}}{{\partial u}}

    \frac{{\partial f}}{{\partial z}} = \frac{{\partial f}}{{\partial v}} \times \frac{{\partial v}}{{\partial x}} + \frac{{\partial f}}{{\partial w}} \times \frac{{\partial w}}{{\partial z}} =  - \frac{{\partial f}}{{\partial v}} + \frac{{\partial f}}{{\partial w}}
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