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I am looking to apply the Implicit Function Theorem to a problem I am working on, and I am unsure if I'm considering the partial derivative correctly. I have a function, say $ h(x,y) = b - x^2 + y$, in which there is an explicit functional relationship $b(x)= 2x^3$. I need to know the partial derivative of $h$ with respect to $x$. My thinking is as follows:

  • If I plugged in $b(x)$, then I'd have $\frac{\partial h}{\partial x} = 6x^2 - 2x $.
  • But, I know in a partial derivative, I'm supposed to ignore dependencies between variables, so I should ignore $\frac{\partial b}{\partial x}$.
  • If I ignore $\frac{\partial b}{\partial x}$, then $\frac{\partial h}{\partial x} = 2x$.
  • But I shouldn't get a different answer of how $h$ depends on $x$, holding $y$ constant, based on whether I call the first term $b$ or $2x^3$.

Is it that partial derivatives only ignore implicit relationships? Or is it because $b$ is not an input to the function?

EDIT: Going to the limit definition of the partial derivative, Does it really just depend on the domain (in the sense of what variables are counted as inputs) of the function? If $b$ is not a separate input to the function, then any change in $b$ due to a change in $x$ is just part of the overall effect of changing $x$. If we were going to allow for $b$ potentially taking on some values other than those given by $b(x)$, then it would make sense to include $b$ an input to the function, but since we only care about $b=b(x)$, that would not make sense.

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  • $\begingroup$ Well, since $b(x)=2x^3$ I do not see any problem if you plug it into $h$. Since $h$ is "lonely", that is, biyective, I would not use the Implicit Function Theorem, I would calculate normally $h_x(x,y)$ and $h_y(x,y)$ (note that $\operatorname{dom}(h)=\Bbb R^2$ and $h\in\mathcal C^1(\Bbb R^2)$). $\endgroup$
    – manooooh
    Oct 23, 2018 at 19:23
  • $\begingroup$ "But, I know in a partial derivative, I'm supposed to ignore dependencies between variables" What do you mean by this? $\endgroup$ Oct 23, 2018 at 19:24
  • $\begingroup$ @MichaelHoppe - I thought that, as in the wikipedia page on partial derivatives, "The difference between the total and partial derivative is the elimination of indirect dependencies between variables in partial derivatives," ie I would ignore any possible $\partial y / \partial x $ term, for example. $\endgroup$ Oct 23, 2018 at 19:32
  • $\begingroup$ @manooooh I'm aware that there's not much need for the implicit function theorem in this example, but the actual function I'm working with would be somewhat tedious to put up. But I still want to know, what is the correct partial derivative? Do I consider that direct dependency, or do I ignore it? $\endgroup$ Oct 23, 2018 at 19:34
  • $\begingroup$ @MathewKnudson so I do not know what is "direct dependency". I guess that it is something like a name of a function; that's it. For example, one could say "If $m(x,y)=n(x)+w(y)$ then what is $m_x(x,y)$, where $n(x)=2x$ and $w(y)=y$?", but I do not see any improve writing like that. $\endgroup$
    – manooooh
    Oct 23, 2018 at 19:38

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