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Math Help - Chain rule with several variables

  1. #1
    MHF Contributor arbolis's Avatar
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    Chain rule with several variables

    Calculate \frac{\partial z}{\partial x} using the chain rule where z=f(u,v,r), u=g(x,y,r), v=h(x,y,r) and r=k(x,y).
    My attempt : \frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\cdot \frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\cdot \frac{\partial v}{\partial x}+\frac{\partial z}{\partial r}\cdot \frac{\partial r}{\partial x}. Is that enough? Or have I to calculate \frac{\partial u}{\partial x} and other terms?
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    Quote Originally Posted by arbolis View Post
    Calculate \frac{\partial z}{\partial x} using the chain rule where z=f(u,v,r), u=g(x,y,r), v=h(x,y,r) and r=k(x,y).
    My attempt : \frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\cdot \frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\cdot \frac{\partial v}{\partial x}+\frac{\partial z}{\partial r}\cdot \frac{\partial r}{\partial x}. Is that enough? Or have I to calculate \frac{\partial u}{\partial x} and other terms?
    Yes --- because

    \frac{\partial u}{\partial x} = \frac{\partial g}{\partial x}+\frac{\partial g}{\partial r} \frac{\partial r}{\partial x}
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