The wikipedia article on chain rule has this example:
Let $u(x,y) =x^2+2y$ where $x(r,t)=rsin(t)$ and $y(r,t)=sin^2(t)$.
Then $$\frac{\partial u}{\partial r} = \frac{\partial u}{\partial x}\frac{\partial x}{\partial r} +\frac{\partial u}{\partial y}\frac{\partial y}{\partial r}$$
I intuitively think of a partial derivative as "derivative assuming all other arguments are fixed). But, in this case, $u$ has two arguments, $x,y$, both of which are changing. So what is the interpretation of the partial of $u$ with respect to $r$?
I get that $u$ is really a function of $r$ and $t$, that we can get by composing $f(a,b) = a^2 +2b$ with $x(r,t)=rsin(t)$ and $y(r,t)=sin^2(t)$, but then why don't we just write $u(r,t)$ instead of $u(x,y)$?
$\partial u / \partial r$ is a very common abuse of notation, which really means $\partial v / \partial r$, where $v$ is the composite function $v(r,t) = u(x(r,t),y(r,t))$.
(Writing $x=x(r,t)$ is also abuse of notation, since the symbol $x$ is used for two different things. It would be more accurate to write $x=\alpha(r,t)$ and $y=\beta(r,t)$ and then $v(r,t) = u(\alpha(r,t),\beta(r,t))$.)