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let $$M^n \subset \mathbb{R}^N$$ $$N^k \subset \mathbb{R}^K$$

be two submanifolds.

We say a function $$f : M \rightarrow N$$ is differentiable if and only if for every map $$(U,\varphi)$$ of M the transformation

$$f \circ \varphi^{-1}: \varphi(U) \subset \mathbb{R}^N \rightarrow \mathbb{R}^K$$

is differentiable.

Does this mean that it is differentiable in the sense of a "normal" function in several variables, thus to say $$d(f \circ \varphi^{-1})(x)=df(\varphi^{-1}(x))d\varphi^{-1}(x)$$?