Why are partial derivatives of functions of two variables themselves functions of two variables?

Question

Why are partial derivatives of functions of two variables themselves functions of two variables?

First, I want to say sorry: I know this is probably a very dumb question... But it's not at all clear to me. Geometrically, it is understandable (considering the slopes of the curves created by intersecting the surface with planes $y=y_o$ or $x=x_o$, it's visually clear that in general the partial derivatives depend on both variables, but I want to know it analytically (considering the definition of partial derivatives as limits). It's not clear to me that the limits (when they exist) generate functions of two variables (or less), unless I consider concrete examples, like $z= sin(x) cos(y)$, compute the partial derivatives and see it. Is there a "proof" or a way to easily see it's true for arbitrary functions? Thanks in advance for the patience with my silly questions...

Answer 1

There is not any particular reason different from what we do in the definition of the derivative funtion for functions of one variable. Indeed, for example, since $f_x(x_0,y_0)$ at a particular point $(x_0,y_0)$ varies in the domain from point to point therefore we can consider the corresponding function and denote that as $f_x(x,y)$.

Answer 2

It depends on your function.

Sometimes your partial derivatives includes terms with both variables and sometimes one or both depend only on one variable. For example for $$ f(x,y)=x^2+y^2$$ we have $$f_x=2x$$ and$$f_y=2y$$

While for $$f(x,y)=x^2y^2$$ we have $$f_x=2xy^2$$, and $$f_y=2x^2y$$