Finding the error in an argument












3












$begingroup$


If $z=f(x,y)$ and $y=x^2$, then by the chain rule



$frac{partial z}{partial x}=frac{partial z}{partial x}frac{partial x}{partial x}+frac{partial z}{partial y}frac{partial y}{partial x}=frac{partial z}{partial x}+2xfrac{partial z}{partial y}$



Therefore



$2xfrac{partial z}{partial y}=0$



and



$frac{partial z}{partial y}=0$



What is wrong with this argument?



I have a feeling that



1.) $x$ and $y$ do not have partial derivatives because they are single-variable, and



2.) $frac{partial z}{partial y}$ cannot be zero, because $y=x^2$ and therefore the derivative of any $y$ term exists.



How is my reasoning? I am pretty confused by this question.






calculus multivariable-calculus partial-derivative







share|cite|improve this question











$endgroup$












  • $begingroup$
    My first question for you is why do you think this line of reasoning is incorrect? Did someone tell you it wasn't right?
    $endgroup$
    – BSplitter
    2 hours ago








  • 1




    $begingroup$
    I also will note that if $2xfrac{partial z}{partial y} =0$, then either $x=0$ or $frac{partial z}{partial y} =0$.
    $endgroup$
    – BSplitter
    2 hours ago










  • $begingroup$
    @BSplitter the problem itself poses this argument as incorrect, the object being to find out why
    $endgroup$
    – mathenthusiast
    2 hours ago
















3












$begingroup$


If $z=f(x,y)$ and $y=x^2$, then by the chain rule



$frac{partial z}{partial x}=frac{partial z}{partial x}frac{partial x}{partial x}+frac{partial z}{partial y}frac{partial y}{partial x}=frac{partial z}{partial x}+2xfrac{partial z}{partial y}$



Therefore



$2xfrac{partial z}{partial y}=0$



and



$frac{partial z}{partial y}=0$



What is wrong with this argument?



I have a feeling that



1.) $x$ and $y$ do not have partial derivatives because they are single-variable, and



2.) $frac{partial z}{partial y}$ cannot be zero, because $y=x^2$ and therefore the derivative of any $y$ term exists.



How is my reasoning? I am pretty confused by this question.






calculus multivariable-calculus partial-derivative







share|cite|improve this question











$endgroup$












  • $begingroup$
    My first question for you is why do you think this line of reasoning is incorrect? Did someone tell you it wasn't right?
    $endgroup$
    – BSplitter
    2 hours ago








  • 1




    $begingroup$
    I also will note that if $2xfrac{partial z}{partial y} =0$, then either $x=0$ or $frac{partial z}{partial y} =0$.
    $endgroup$
    – BSplitter
    2 hours ago










  • $begingroup$
    @BSplitter the problem itself poses this argument as incorrect, the object being to find out why
    $endgroup$
    – mathenthusiast
    2 hours ago














3












3








3





$begingroup$


If $z=f(x,y)$ and $y=x^2$, then by the chain rule



$frac{partial z}{partial x}=frac{partial z}{partial x}frac{partial x}{partial x}+frac{partial z}{partial y}frac{partial y}{partial x}=frac{partial z}{partial x}+2xfrac{partial z}{partial y}$



Therefore



$2xfrac{partial z}{partial y}=0$



and



$frac{partial z}{partial y}=0$



What is wrong with this argument?



I have a feeling that



1.) $x$ and $y$ do not have partial derivatives because they are single-variable, and



2.) $frac{partial z}{partial y}$ cannot be zero, because $y=x^2$ and therefore the derivative of any $y$ term exists.



How is my reasoning? I am pretty confused by this question.






calculus multivariable-calculus partial-derivative







share|cite|improve this question











$endgroup$




If $z=f(x,y)$ and $y=x^2$, then by the chain rule



$frac{partial z}{partial x}=frac{partial z}{partial x}frac{partial x}{partial x}+frac{partial z}{partial y}frac{partial y}{partial x}=frac{partial z}{partial x}+2xfrac{partial z}{partial y}$



Therefore



$2xfrac{partial z}{partial y}=0$



and



$frac{partial z}{partial y}=0$



What is wrong with this argument?



I have a feeling that



1.) $x$ and $y$ do not have partial derivatives because they are single-variable, and



2.) $frac{partial z}{partial y}$ cannot be zero, because $y=x^2$ and therefore the derivative of any $y$ term exists.



How is my reasoning? I am pretty confused by this question.






calculus multivariable-calculus partial-derivative




calculus multivariable-calculus partial-derivative






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 3 hours ago







mathenthusiast

















asked 3 hours ago









mathenthusiastmathenthusiast

758




758












  • $begingroup$
    My first question for you is why do you think this line of reasoning is incorrect? Did someone tell you it wasn't right?
    $endgroup$
    – BSplitter
    2 hours ago








  • 1




    $begingroup$
    I also will note that if $2xfrac{partial z}{partial y} =0$, then either $x=0$ or $frac{partial z}{partial y} =0$.
    $endgroup$
    – BSplitter
    2 hours ago










  • $begingroup$
    @BSplitter the problem itself poses this argument as incorrect, the object being to find out why
    $endgroup$
    – mathenthusiast
    2 hours ago


















  • $begingroup$
    My first question for you is why do you think this line of reasoning is incorrect? Did someone tell you it wasn't right?
    $endgroup$
    – BSplitter
    2 hours ago








  • 1




    $begingroup$
    I also will note that if $2xfrac{partial z}{partial y} =0$, then either $x=0$ or $frac{partial z}{partial y} =0$.
    $endgroup$
    – BSplitter
    2 hours ago










  • $begingroup$
    @BSplitter the problem itself poses this argument as incorrect, the object being to find out why
    $endgroup$
    – mathenthusiast
    2 hours ago
















$begingroup$
My first question for you is why do you think this line of reasoning is incorrect? Did someone tell you it wasn't right?
$endgroup$
– BSplitter
2 hours ago






$begingroup$
My first question for you is why do you think this line of reasoning is incorrect? Did someone tell you it wasn't right?
$endgroup$
– BSplitter
2 hours ago






1




1




$begingroup$
I also will note that if $2xfrac{partial z}{partial y} =0$, then either $x=0$ or $frac{partial z}{partial y} =0$.
$endgroup$
– BSplitter
2 hours ago




$begingroup$
I also will note that if $2xfrac{partial z}{partial y} =0$, then either $x=0$ or $frac{partial z}{partial y} =0$.
$endgroup$
– BSplitter
2 hours ago












$begingroup$
@BSplitter the problem itself poses this argument as incorrect, the object being to find out why
$endgroup$
– mathenthusiast
2 hours ago




$begingroup$
@BSplitter the problem itself poses this argument as incorrect, the object being to find out why
$endgroup$
– mathenthusiast
2 hours ago










1 Answer
1






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oldest

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4












$begingroup$

Nothing wrong. Just change it into



$$frac{d z}{d x}=frac{partial z}{partial x}frac{partial x}{partial x}+frac{partial z}{partial y}frac{partial y}{partial x}=frac{partial z}{partial x}+2xfrac{partial z}{partial y}$$



Note that that the first term is $frac{d z}{d x}$, which is different from $frac{partial z}{partial x}$. So they cannot cancel out. The partial derivatives do exist, but be careful not to mix it up with $frac{d z}{d x}$, which is NOT a partial derivative.



Actually, a better way to say this is that



$$left[frac{partial z}{partial x}right]_{y=x^2}=frac{partial z}{partial x}frac{partial x}{partial x}+frac{partial z}{partial y}frac{partial y}{partial x}=frac{partial z}{partial x}+2xfrac{partial z}{partial y}.$$



Where I have clearly written down the restriction $y=x^2$.






share|cite|improve this answer









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    4












    $begingroup$

    Nothing wrong. Just change it into



    $$frac{d z}{d x}=frac{partial z}{partial x}frac{partial x}{partial x}+frac{partial z}{partial y}frac{partial y}{partial x}=frac{partial z}{partial x}+2xfrac{partial z}{partial y}$$



    Note that that the first term is $frac{d z}{d x}$, which is different from $frac{partial z}{partial x}$. So they cannot cancel out. The partial derivatives do exist, but be careful not to mix it up with $frac{d z}{d x}$, which is NOT a partial derivative.



    Actually, a better way to say this is that



    $$left[frac{partial z}{partial x}right]_{y=x^2}=frac{partial z}{partial x}frac{partial x}{partial x}+frac{partial z}{partial y}frac{partial y}{partial x}=frac{partial z}{partial x}+2xfrac{partial z}{partial y}.$$



    Where I have clearly written down the restriction $y=x^2$.






    share|cite|improve this answer









    $endgroup$


















      4












      $begingroup$

      Nothing wrong. Just change it into



      $$frac{d z}{d x}=frac{partial z}{partial x}frac{partial x}{partial x}+frac{partial z}{partial y}frac{partial y}{partial x}=frac{partial z}{partial x}+2xfrac{partial z}{partial y}$$



      Note that that the first term is $frac{d z}{d x}$, which is different from $frac{partial z}{partial x}$. So they cannot cancel out. The partial derivatives do exist, but be careful not to mix it up with $frac{d z}{d x}$, which is NOT a partial derivative.



      Actually, a better way to say this is that



      $$left[frac{partial z}{partial x}right]_{y=x^2}=frac{partial z}{partial x}frac{partial x}{partial x}+frac{partial z}{partial y}frac{partial y}{partial x}=frac{partial z}{partial x}+2xfrac{partial z}{partial y}.$$



      Where I have clearly written down the restriction $y=x^2$.






      share|cite|improve this answer









      $endgroup$
















        4












        4








        4





        $begingroup$

        Nothing wrong. Just change it into



        $$frac{d z}{d x}=frac{partial z}{partial x}frac{partial x}{partial x}+frac{partial z}{partial y}frac{partial y}{partial x}=frac{partial z}{partial x}+2xfrac{partial z}{partial y}$$



        Note that that the first term is $frac{d z}{d x}$, which is different from $frac{partial z}{partial x}$. So they cannot cancel out. The partial derivatives do exist, but be careful not to mix it up with $frac{d z}{d x}$, which is NOT a partial derivative.



        Actually, a better way to say this is that



        $$left[frac{partial z}{partial x}right]_{y=x^2}=frac{partial z}{partial x}frac{partial x}{partial x}+frac{partial z}{partial y}frac{partial y}{partial x}=frac{partial z}{partial x}+2xfrac{partial z}{partial y}.$$



        Where I have clearly written down the restriction $y=x^2$.






        share|cite|improve this answer









        $endgroup$



        Nothing wrong. Just change it into



        $$frac{d z}{d x}=frac{partial z}{partial x}frac{partial x}{partial x}+frac{partial z}{partial y}frac{partial y}{partial x}=frac{partial z}{partial x}+2xfrac{partial z}{partial y}$$



        Note that that the first term is $frac{d z}{d x}$, which is different from $frac{partial z}{partial x}$. So they cannot cancel out. The partial derivatives do exist, but be careful not to mix it up with $frac{d z}{d x}$, which is NOT a partial derivative.



        Actually, a better way to say this is that



        $$left[frac{partial z}{partial x}right]_{y=x^2}=frac{partial z}{partial x}frac{partial x}{partial x}+frac{partial z}{partial y}frac{partial y}{partial x}=frac{partial z}{partial x}+2xfrac{partial z}{partial y}.$$



        Where I have clearly written down the restriction $y=x^2$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 2 hours ago









        Holding ArthurHolding Arthur

        1,360417




        1,360417






























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