# Notation in differential map between differential manifolds

When we define a differential map on $$x \in M$$ as an aplication between differential manifolds $$f : M \to N$$ such that $$d(\psi \circ f \circ \phi^{-1})_{\phi(x)} : \mathbb{R}^m \to \mathbb{R}^n$$ is differential in usual sense.

My question is if we can remove $$\phi(x)$$ in the definition of $$d(\psi \circ f \circ \phi^{-1})_{\phi(x)}$$ because you can pass directly to $$d(\psi \circ f)(x)$$, yeah? Or is it simple notation and you ever must write it?

The lower $$\phi(x)$$ is a subscript, and it is not part of a composition against the "neighbor" $$\phi^{-1}$$ that may cancel. It is a location marker, telling you where you are taking the real derivative.