How to anti-derivative the partial derivative of a multivariate function across all variables to end up with...












0












$begingroup$


Partial differentiation of the function



$$y=hat{y} exp{left(-sum^n_{i=1}frac{alpha_i}{x_i}right)}$$



with respect to $x_i$ gives



$$frac{partial y}{partial x_i}=yfrac{alpha_i}{x_i^2}$$



which can be rearranged



$$frac{partial ln y}{partial ln x_i}=alpha_ifrac{d ln x_i}{dx_i}$$



(You see physicists doing this sort of thing anyway.)



How can I reverse this process to start with the last equation and end up with the first? If you have major objections to the rearrangement in the third equation, then how can I start with the second equation and end up with the first?






calculus integration exponential-function partial-derivative







share|cite|improve this question









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    0












    $begingroup$


    Partial differentiation of the function



    $$y=hat{y} exp{left(-sum^n_{i=1}frac{alpha_i}{x_i}right)}$$



    with respect to $x_i$ gives



    $$frac{partial y}{partial x_i}=yfrac{alpha_i}{x_i^2}$$



    which can be rearranged



    $$frac{partial ln y}{partial ln x_i}=alpha_ifrac{d ln x_i}{dx_i}$$



    (You see physicists doing this sort of thing anyway.)



    How can I reverse this process to start with the last equation and end up with the first? If you have major objections to the rearrangement in the third equation, then how can I start with the second equation and end up with the first?






    calculus integration exponential-function partial-derivative







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Partial differentiation of the function



      $$y=hat{y} exp{left(-sum^n_{i=1}frac{alpha_i}{x_i}right)}$$



      with respect to $x_i$ gives



      $$frac{partial y}{partial x_i}=yfrac{alpha_i}{x_i^2}$$



      which can be rearranged



      $$frac{partial ln y}{partial ln x_i}=alpha_ifrac{d ln x_i}{dx_i}$$



      (You see physicists doing this sort of thing anyway.)



      How can I reverse this process to start with the last equation and end up with the first? If you have major objections to the rearrangement in the third equation, then how can I start with the second equation and end up with the first?






      calculus integration exponential-function partial-derivative







      share|cite|improve this question









      $endgroup$




      Partial differentiation of the function



      $$y=hat{y} exp{left(-sum^n_{i=1}frac{alpha_i}{x_i}right)}$$



      with respect to $x_i$ gives



      $$frac{partial y}{partial x_i}=yfrac{alpha_i}{x_i^2}$$



      which can be rearranged



      $$frac{partial ln y}{partial ln x_i}=alpha_ifrac{d ln x_i}{dx_i}$$



      (You see physicists doing this sort of thing anyway.)



      How can I reverse this process to start with the last equation and end up with the first? If you have major objections to the rearrangement in the third equation, then how can I start with the second equation and end up with the first?






      calculus integration exponential-function partial-derivative




      calculus integration exponential-function partial-derivative






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 14 at 13:22









      benben

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      95111129






















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