Why do we need charts to define a submersion/immersion?

I am learning the theory of smooth manifolds and have a question on the definitions of a submersion/immersion and its dependency on given charts.

Given a smooth map $$f:M\mapsto N$$ between two smooth manifolds of finite dimension. If I am correct this means that given any chart $$\chi$$ of $$M$$ and chart $$\chi^\prime$$ on $$N$$, $$f_{\chi^\prime}^{\chi}=\chi^\prime\circ f\circ\chi^{-1}$$ is smooth in the usual sense of analysis.

Now to prove if $$f$$ is a submersion (or similar an immersion) at $$p\in M$$ one checks that, $$(df_{\chi^\prime}^{\chi})_{\chi(p)}$$ is surjective\injective. By the chainrule, $$\big(d(\chi^\prime\circ f\circ\chi^{-1})\big)_{\chi(p)}=(d\chi^\prime)_{\chi(p)}\circ(df)_p\circ(d\chi^{-1})_{\chi^{-1}(p)}.$$ But since all charts a homeomorphisms their differentials are isomorfisms.

Now my question is, why bother looking at $$f_{\chi^\prime}^{\chi}$$ if you can just look at whether or not the differential of $$f$$ is surjective/injecitive? The differentials of the charts are after all isomorfisms. Am i looking at it the right way?

Beside that, in the practical situation of having to check wheter or not a map is a submersion/immersion one has to do this for all combination of charts contained in the two atlases which induce the smooth structures, thats a bit cumbersome... Is there a trick/theorem one can use?

To put it simply, you can. You ise an atlas to define a smooth structure, so the fact that $$f:M\to N$$ is smooth is in part determined by the coordinate atlases, but for any smooth function, $$df_p$$ is a coordinate-independent linear map from $$T_pM\to T_{f(p)}N$$, and immersions/submersions are typically defined in terms of the rank of $$df$$ without reference to coordinates.
Specifically, we say $$f$$ is
• an immersion if $$df_p$$ is injective at each $$p\in m$$
• a submersion if $$df_p$$ is surjective at each $$p\in m$$
As you have shown, the rank of $$df$$ is equal to the rank of its local representative in any coordinates, so we can check if a map is a submersion/immersion in coordinates without issue. There's no need to use every combination of charts, only enough to cover each point in $$M$$ at least once.