There are two distinct meanings of taking the derivative of a function ##f(x,y)## "with respect to x".
These are
Case 1) Taking the partial derivative of ##f## with respect to its
first argument, which has been named "x" in this example.
Case 2) Taking the dervative of ##f## with respect to the variable ##x##
wherever ##x## is involved.
For example, suppose we are given
##f(x,y) = x^2 + y## and ##y = 3x^3##.
case 1) ##\frac{\partial f}{\partial x} = 2x##
case 2) ##\frac{df}{dx} = 2x + 9x^2##
These two distinct concepts are a perpetual source of confusion in reading material written by physicists. (See
@andrewkirk 's
https://www.physicsforums.com/insights/partial-differentiation-without-tears/ )
To avoid such confusion, some authors have gone so far as using notation like ##f_1(x,y)## when they refer to the derivative of ##f## with respect to its first argument. This is a more straightforward notation that traditional notation like ##\frac{\partial f}{\partial w}## because to know the meaning of that notation, you must know which position ##w## occupies in the list of arguments of ##f##.
The functions defined by ##f(w,r) = w^2 + 3r ## and ##f(r,w) = r^2 + 3w ## are the the same function unless other information has been given to distinguish ##w## and ##r##. The partial derivative of ##f## with repspect to its first argument can be denoted as ##f_1(x,y) = 2x## or ##f_1(w,r) = 2w## or ##f_1(r,w) = 2r##. Technically variable names are abitrary. In applying math, a variable talked about on one page may have a special meaning that carries over to the next page, so people don't exercise their full freedom of choice when defining functions. Also there are traditions such as using "##x##" to be the first argument of a function.