Ι have a question concerning symbolization for derivatives when making a substitution.
For instance, let $f$ be a two variable function, where $$ f(x,y)=x^2+y \quad , \; (x,y) \in \mathbb{R^2} $$ Then, $$ \frac{\partial f}{\partial x}(x,y)=2x $$ If we want to make a substitution, i.e. replace $x$ with another function $u(x)$ should we write $$ \frac{\partial f}{\partial x} \bigl (u(x),y \bigr )=2u(x) $$ or $$ \frac{\partial f}{\partial u} \bigl (u(x),y \bigr )=2u(x) $$
So, should we also replace $ \partial x $ with $ \partial u $ ?
If the first is true, doesn't it contradict the chain rule ?
Thank you in advance