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I was going over the answers in this question and was wondering:

  1. In the case of a function defined by $f(x(u,v), y(u,v))$, are $\partial f/\partial x$ and $\partial f/\partial y$ computed in the "normal" way where we just treat $x$ and $y$ as regular variables and not functions?
  2. What happens if $x(u,v)$ and $y(u,v)$ are such that $y$ can be expressed in terms of $x$ aka $y=y(x)$? In this case is $\frac{\partial f(x,y(x))}{\partial y}$ the same as in the above where $x$ is constant or is $\frac{\partial f(x,y(x))}{\partial y}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial y}+\frac{\partial f}{\partial y}$ since if $y=y(x)$ then it's possible $x=x(y)$? Similarly, is $\frac{\partial f(x,y(x))}{\partial x}=\frac{\partial f}{\partial y}\frac{\partial y}{\partial x}+\frac{\partial f}{\partial x}$ or just $\frac{\partial f}{\partial x}$?

In general, how do I interpret $\partial f/\partial x$ and $\partial f/\partial y$ in either case, given that you likely can't alter $x$ without altering $y$ and vice versa?

Yandle
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1 Answers1

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  1. Yes, $\partial f/ \partial x$ measures how $f$ changes when we change its first argument $x$. In this case, the first argument happens to be a function of two other quantities $u$ and $v$.

  2. Suppose $y = y(x)$ is a function of $x$. Then by (1) the $y$-partial $\partial f/ \partial y$ is taken "normally" as if $y$ is an independent variable. However, when computing $\partial f/ \partial x$ we have to use the chain rule, since changing $x$ causes both of $f$'s arguments to change.

The $x$-partial works out to $$ \frac{\partial f(x, y(x))}{\partial x} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial x} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial x} = \frac{\partial f}{\partial x} \cdot 1 + \frac{\partial f}{\partial y}\frac{\partial y}{\partial x}.$$

To summarize: the quantities $\partial f / \partial x$ and $\partial f/ \partial y$ measure how $f$ changes under small perturbations of its first and second argument, respectively. The product term in the chain rule accounts for when the first argument inadvertently changes the second argument.

Sam Freedman
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