In Milnor's Topology from a differentiable viewpoint on page 1, functions on open subsets of $\mathbb{R}^k$ to $\mathbb{R}^l$ are defined to be smooth if their partial derivatives exist and are continuous. Then he defines smooth manifolds as subsets of $\mathbb{R}^k$ (which I understand is equivalent to the abstract definition of manifolds due to Whitney's theorem).
My question is that while defining functions on a smooth manifold $M$, is it correct to say that the euclidean space $M$ is embedded in is assumed to have the standard smooth structure?
If so, what happens when that is not the case? For example consider the manifold $$M \subset \mathbb{R}, M = (-1,1)$$ with an atlas containing the single chart $$\phi : x \mapsto x^{1/9}$$ and define $f : M \rightarrow \mathbb{R}, f(x) = x^{1/3}$
Now $f$ is smooth because $f \circ \phi^{-1} : \mathbb{R} \rightarrow \mathbb{R}$ is smooth.
The problem arises when on page 6, derivative of a smooth function on manifolds is defined to be $$df_x(h) = \lim_{t \rightarrow 0} (f(x + th) - f(x))/t$$ the limit of which in our case doesn't exist at $x = 0$.
To be precise the actual derivative is the derivative of an extention of $f$ on open subsets of $M$ but in this case the two are the same.
Yes, $\mathbb{R}^k$ is always considered to have the standard smooth structure in this context. If you have a manifold embedded in $\mathbb{R}^k$ with respect to some nonstandard smooth structure, then it is just not a manifold in Milnor's sense and his definitions do not apply. You would instead need to use the general theory of abstract manifolds.