Chain rule - Find $frac{partial z}{partial x}$ and $frac{partial z}{partial y} $ if $z=pq+qw$












0












$begingroup$



Find $frac{partial z}{partial x}$ and $frac{partial z}{partial y} $ if



$z=pq+qw$



$p=2x-y$, $q=x-2y$ and $w=-2x-2y$




Is $frac{partial z}{partial x}$ equals to:



$frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
frac{partial z}{partial q} frac{partial q}{partial x}+ frac{partial z}{partial w} frac{partial w}{partial x}$



Or
$frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
frac{partial z}{partial w} frac{partial w}{partial x}$



I can complete the solution, I need only the correct rule. Thanks






calculus chain-rule







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    0












    $begingroup$



    Find $frac{partial z}{partial x}$ and $frac{partial z}{partial y} $ if



    $z=pq+qw$



    $p=2x-y$, $q=x-2y$ and $w=-2x-2y$




    Is $frac{partial z}{partial x}$ equals to:



    $frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
    frac{partial z}{partial q} frac{partial q}{partial x}+ frac{partial z}{partial w} frac{partial w}{partial x}$



    Or
    $frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
    frac{partial z}{partial w} frac{partial w}{partial x}$



    I can complete the solution, I need only the correct rule. Thanks






    calculus chain-rule







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$



      Find $frac{partial z}{partial x}$ and $frac{partial z}{partial y} $ if



      $z=pq+qw$



      $p=2x-y$, $q=x-2y$ and $w=-2x-2y$




      Is $frac{partial z}{partial x}$ equals to:



      $frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
      frac{partial z}{partial q} frac{partial q}{partial x}+ frac{partial z}{partial w} frac{partial w}{partial x}$



      Or
      $frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
      frac{partial z}{partial w} frac{partial w}{partial x}$



      I can complete the solution, I need only the correct rule. Thanks






      calculus chain-rule







      share|cite|improve this question









      $endgroup$





      Find $frac{partial z}{partial x}$ and $frac{partial z}{partial y} $ if



      $z=pq+qw$



      $p=2x-y$, $q=x-2y$ and $w=-2x-2y$




      Is $frac{partial z}{partial x}$ equals to:



      $frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
      frac{partial z}{partial q} frac{partial q}{partial x}+ frac{partial z}{partial w} frac{partial w}{partial x}$



      Or
      $frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
      frac{partial z}{partial w} frac{partial w}{partial x}$



      I can complete the solution, I need only the correct rule. Thanks






      calculus chain-rule




      calculus chain-rule






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      share|cite|improve this question











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      asked Dec 3 '18 at 17:32









      DimaDima

      615416




      615416






















          2 Answers
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          1












          $begingroup$

          $$frac{partial z}{partial x}=frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
          frac{partial z}{partial w} frac{partial w}{partial x}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you so much
            $endgroup$
            – Dima
            Dec 3 '18 at 20:00



















          1












          $begingroup$

          $frac{partial z}{partial x}=frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
          frac{partial z}{partial w} frac{partial w}{partial x}=$          , then



          $frac{partial z}{partial x}=p+w$ and



          $frac{partial z}{partial y}=-3q-2p-2w$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you so much, for $frac{partial z}{partial y}$ I think it is $-q-2(p+w)$, what do you think?
            $endgroup$
            – Dima
            Dec 3 '18 at 19:59










          • $begingroup$
            Thankful. no $frac{partial z}{partial w}frac{partial w}{partial y}=-2q$
            $endgroup$
            – yavar
            Dec 4 '18 at 13:58











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          2 Answers
          2






          active

          oldest

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          2 Answers
          2






          active

          oldest

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          active

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          active

          oldest

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          1












          $begingroup$

          $$frac{partial z}{partial x}=frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
          frac{partial z}{partial w} frac{partial w}{partial x}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you so much
            $endgroup$
            – Dima
            Dec 3 '18 at 20:00
















          1












          $begingroup$

          $$frac{partial z}{partial x}=frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
          frac{partial z}{partial w} frac{partial w}{partial x}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you so much
            $endgroup$
            – Dima
            Dec 3 '18 at 20:00














          1












          1








          1





          $begingroup$

          $$frac{partial z}{partial x}=frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
          frac{partial z}{partial w} frac{partial w}{partial x}$$






          share|cite|improve this answer









          $endgroup$



          $$frac{partial z}{partial x}=frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
          frac{partial z}{partial w} frac{partial w}{partial x}$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 3 '18 at 17:37









          Anubhab GhosalAnubhab Ghosal

          81618




          81618












          • $begingroup$
            Thank you so much
            $endgroup$
            – Dima
            Dec 3 '18 at 20:00


















          • $begingroup$
            Thank you so much
            $endgroup$
            – Dima
            Dec 3 '18 at 20:00
















          $begingroup$
          Thank you so much
          $endgroup$
          – Dima
          Dec 3 '18 at 20:00




          $begingroup$
          Thank you so much
          $endgroup$
          – Dima
          Dec 3 '18 at 20:00











          1












          $begingroup$

          $frac{partial z}{partial x}=frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
          frac{partial z}{partial w} frac{partial w}{partial x}=$          , then



          $frac{partial z}{partial x}=p+w$ and



          $frac{partial z}{partial y}=-3q-2p-2w$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you so much, for $frac{partial z}{partial y}$ I think it is $-q-2(p+w)$, what do you think?
            $endgroup$
            – Dima
            Dec 3 '18 at 19:59










          • $begingroup$
            Thankful. no $frac{partial z}{partial w}frac{partial w}{partial y}=-2q$
            $endgroup$
            – yavar
            Dec 4 '18 at 13:58
















          1












          $begingroup$

          $frac{partial z}{partial x}=frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
          frac{partial z}{partial w} frac{partial w}{partial x}=$          , then



          $frac{partial z}{partial x}=p+w$ and



          $frac{partial z}{partial y}=-3q-2p-2w$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you so much, for $frac{partial z}{partial y}$ I think it is $-q-2(p+w)$, what do you think?
            $endgroup$
            – Dima
            Dec 3 '18 at 19:59










          • $begingroup$
            Thankful. no $frac{partial z}{partial w}frac{partial w}{partial y}=-2q$
            $endgroup$
            – yavar
            Dec 4 '18 at 13:58














          1












          1








          1





          $begingroup$

          $frac{partial z}{partial x}=frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
          frac{partial z}{partial w} frac{partial w}{partial x}=$          , then



          $frac{partial z}{partial x}=p+w$ and



          $frac{partial z}{partial y}=-3q-2p-2w$






          share|cite|improve this answer









          $endgroup$



          $frac{partial z}{partial x}=frac{partial z}{partial p} frac{partial p}{partial x}+ frac{partial z}{partial q} frac{partial q}{partial x}+
          frac{partial z}{partial w} frac{partial w}{partial x}=$          , then



          $frac{partial z}{partial x}=p+w$ and



          $frac{partial z}{partial y}=-3q-2p-2w$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 3 '18 at 19:57









          yavaryavar

          843




          843












          • $begingroup$
            Thank you so much, for $frac{partial z}{partial y}$ I think it is $-q-2(p+w)$, what do you think?
            $endgroup$
            – Dima
            Dec 3 '18 at 19:59










          • $begingroup$
            Thankful. no $frac{partial z}{partial w}frac{partial w}{partial y}=-2q$
            $endgroup$
            – yavar
            Dec 4 '18 at 13:58


















          • $begingroup$
            Thank you so much, for $frac{partial z}{partial y}$ I think it is $-q-2(p+w)$, what do you think?
            $endgroup$
            – Dima
            Dec 3 '18 at 19:59










          • $begingroup$
            Thankful. no $frac{partial z}{partial w}frac{partial w}{partial y}=-2q$
            $endgroup$
            – yavar
            Dec 4 '18 at 13:58
















          $begingroup$
          Thank you so much, for $frac{partial z}{partial y}$ I think it is $-q-2(p+w)$, what do you think?
          $endgroup$
          – Dima
          Dec 3 '18 at 19:59




          $begingroup$
          Thank you so much, for $frac{partial z}{partial y}$ I think it is $-q-2(p+w)$, what do you think?
          $endgroup$
          – Dima
          Dec 3 '18 at 19:59












          $begingroup$
          Thankful. no $frac{partial z}{partial w}frac{partial w}{partial y}=-2q$
          $endgroup$
          – yavar
          Dec 4 '18 at 13:58




          $begingroup$
          Thankful. no $frac{partial z}{partial w}frac{partial w}{partial y}=-2q$
          $endgroup$
          – yavar
          Dec 4 '18 at 13:58


















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