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Math Help - Partial derivative problem

  1. #1
    Junior Member
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    Partial derivative problem

    If u=\frac{xy}{z}lnx+xf(\frac{y}{x},\frac{z}{x}) and f is differentiable at least once, prove that:

    x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z}=u+\frac{xy}{z}

    How to find the partial derivative with respect of x,y,z of that function f when it's not expressed.

    Thanks
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  2. #2
    MHF Contributor
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    Re: Partial derivative problem

    Just in case a picture helps...



    ... where (key in spoiler) ...

    Spoiler:


    is the chain rule for two inner functions, i.e...

    \frac{d}{dx}\ f(v(x), w(x)) = \frac{\partial f}{\partial v} \frac{dv}{dx} + \frac{\partial f}{\partial w} \frac{dw}{dx}

    As with...



    ... the ordinary chain rule, straight continuous lines differentiate downwards (integrate up) with respect to x (or whatever), and the straight dashed line similarly but with respect to the (corresponding) dashed balloon expression which is (one of) the inner function(s) of the composite expression.





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