Notation: The appropriate way to write the derivative of a function with respect to a variable it doesn't depend on

Question

If I have some function $f$ that is purely a function of $u$ and $v$, it would be $f(u,v)$. If I want to take the derivative of the function with respect to say, $z$, should I re-write this function as $f(u,v,z)$ and the derivative as $\partial f(u,v,z)/\partial z$? Or is $\partial f(u,v)/\partial z$ acceptable? I am thinking the former since for a constant function $f(x) = 5$ where $df/dx = 0$, we still have the variable in there even if the function doesn't depend on it.

Answer 1

If you intend to take derivative with respect to $z$ you include $z$ in your function. For example $$f(x,y,z)= x^2+y^2-10$$ is a function and we have $$ \partial f(x,y,z)/\partial z=0.$$

The benefit of including $z$ in the definition of function is to show that $z$ is independent of $x$ and $y$