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Derivative of a function of two variables w.r.t. one variable

I’m in the process of learning about functions of several variables. My question is that, if a function $f(x_1,x_2,\ldots,x_n)$ depends on multiple variables how do I find its derivative with respect to one of those variables?
For example, I want to find the derivative of
$f(x,y) = xy$
with respect to $x$. I know there is a simple way of doing it by substituting $x$ for $x_1$ and using the chain rule, but I was wondering if there is a more general way that allows me to differentiate in any way, e.g. with respect to $y$, using the chain rule.

A:

If you want to differentiate a function with respect to a given variable, you are actually defining the partial derivative of that function.
From what you want to derive, it seems that you’re looking for the gradients, however there is already a comprehensive introduction to gradients over there.
In particular, $y$ is the only variable, and we want to find the derivative of $xy$ with respect to $y$. Let’s denote this derivative by $Df(y)$, and write the function as $f(x,y) = xy$. To make it explicit, the derivative becomes
$$
Df(y) = y\cdot Df(x,y) = \frac{1}{1

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