I'm studying derivatives 101 and I can't get my head around the phrasing "with respect to" something.
Eg in chain rule we calculate the derivative of outer function with respect to inner + derivative of inner with respect to x. But what does it actually mean (in human language) to say a derivative is "with respect to" anything at all?
Thanks a ton.
Best Answer
If a function depends on only one variable, then its derivative is of course 'with respect to' that one variable, because the function only depends on one parameter, so there is no need to distinguish which parameter we are talking about.
But if it depends on two variables it is slightly more clear. For $f(x,y)$, the derivative with respect to $x$, is $\frac{df}{dx}$ and the derivative with respect to $y$ is $\frac{df}{dy}$. So if we let $$ f(x,y) = x + y^2 \\ \frac{\partial f}{\partial x} = 1 \\ \frac{\partial f}{\partial y} = 2 y $$ we can see these quantities are not the same. The derivative with respect to $x$ is: "at what rate does $f$ change as $x$ changes", in this case it is a constant, $1$. At what rate does $f$ change as $y$ changes, i.e. "the derivative with respect to $y$", which goes like $2y$.
I hope that is what you are looking for.
Note: Hurkyl's comments below are very important, in this instance we have to use a slightly different notation $\partial$ for the derivative where there is more than one parameter, because there may be co-dependence between parameters. I had originally intended to keep the explanation simple as you had indicated it was 'derivatives 101'.