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When you take the derivative of an expression with respect to x, does x have to be a variable, or is it allowed to be a polynomial, a term, a vector, or anything else? It doesn't seem to make sense to me if x is not a variable.

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Could you please elaborate on the context in which this question arises? A thorough answer to the question as it stands would have to cover a lot of ground, so it would help to have a better idea of what to focus on. Also, what do you mean by "term"? – Jonas Meyer Jan 18 '11 at 1:38
Also, what do you mean by "variable"? – Qiaochu Yuan Jan 18 '11 at 2:14
@Jonas well, just generally in calculus. By "term" I mean some expressions multiplied together without pluses or minuses. – wrongusername Jan 18 '11 at 2:19
@Qiaochu by "variable" I mean like a single symbol that represents an unknown quantity – wrongusername Jan 18 '11 at 2:20
@wrongusername: well, then polynomials and vectors can also be variables. – Qiaochu Yuan Jan 18 '11 at 2:22
up vote 5 down vote accepted

Intuitively, the derivative of $f$ with respect to $u$ is the limit of the change in $f$ as $u$ changes, divided by the change in $u$, as the change in $u$ vanishes. This does not require $u$ to be a "variable" in the usual sense: you can certainly ask for the rate of change of, say, $f(x) = \sin(x^2+1)$ with respect to $u=x^2$. So, no, it does not have to be an "independent variable" in the sense that you seem to be thinking about.

In fact, that's what the Chain Rule is all about! It tells you that if $f$ depends on $g$ and $g$ depends on $x$, then the rate of change of $f$ with respect to $x$ is equal to the rate of change of $f$ with respect to $g$, times the rate of change of $g$ with respect to $x$: $$\frac{df}{dx} = \frac{df}{dg}\;\frac{dg}{dx}.$$ Here, we usually have $g$ a function, not a "variable". Yet we can talk about the derivative of $f$ with respect to $g$.

Every time you have a function, you can try to talk about the rate of change of the function with respect to something else, provided you have some way of quantifying the change.

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Thank you! with respect to . With respect to what? – wrongusername Jan 18 '11 at 8:08
@wrongusername: huh? All of my "with respect to"s seem to have a clause following it. First one is with respect to $u$; next is $x$; then $g$; then $x$; and finally "to something else". – Arturo Magidin Jan 18 '11 at 13:58
There seems to be a LaTeX rendering problem. Before your first paragraph, I'm seeing a "$u=x^2$" that I'm certain belongs after that paragraph's "with respect to" (which, on my --and, presumably, @wrongusername's-- screen, looks to be followed only by "."). All of your other in-line and displayed LaTeX expressions appear as expected. I've seen this problem a couple of times lately (in Safari 5.0.3 on Mac OS X 10.6.6). – Blue Jan 18 '11 at 14:49
@Day Late Don: Ah; I see no problem on Firefox. Has this been brought up in meta? You may also try a hard refresh. – Arturo Magidin Jan 18 '11 at 14:54

You could look at matrix calculus for example.

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