I heard the following:
For a smooth submanifold $M \subseteq \mathbb{R}^n$, given a local trivialization one can easily find a basis for the tangent space.
I want to know how. So first I should maybe formalize the question: Suppose $M \subseteq \mathbb{R}^n$ is a smooth $k$-dimensional submanifold and for $p \in M$,$\phi:U \rightarrow V$ a local trivialization of $M$ such that $x \in M \cap U$. Then there exist functions $b_1,...,b_k$ such that $B=\{b_1(x),...,b_k(x)\}$ form a basis of $T_xM$.
My idea: If $\phi:U \rightarrow V$ is a local trivialization, then $\frac{\partial}{\partial x_1} \phi(x)_{|x=p},...,\frac{\partial}{\partial x_k}\phi(x)_{|x=p}$ should be a basis for the tangent space $T_pM$. My problems at the moment: I don't know how to explicitly show my assumption. I.e. I need to show that for $v \in T_pM$, $v$ can be written as a linear combination of $\frac{\partial}{\partial x_1} \phi(x)_{|x=p},...,\frac{\partial}{\partial x_k}\phi(x)_{|x=p}$. And I also need to show that $\frac{\partial}{\partial x_1} \phi(x)_{|x=p},...,\frac{\partial}{\partial x_k}\phi(x)_{|x=p}$ are linearly independent.
Question: How do I show above-mentioned? Does my idea work or am I wrong?
Edit:
I wanted to add the Definitions I use: Let $M \subseteq \mathbb{R}^n$. $M$ is a smooth $k$-dimensional submanifold if for each $x \in M$, there exists open subsets $U,V \subseteq \mathbb{R}^n$ were $x \in U$ and there exists a diffeomorphism $\phi:U\rightarrow V$ such that $\phi(U \cap M) = V \cap \mathbb{R}^k$. I call $\phi$ a local trivialization. As far as I know, $\phi$ is sometimes also called a local parametrization.
I want to show that there exist local coordinates $b_1(x),...,b_k(x)$ of the tangent space $T_xM$.
You are making things appear more complicated than they are by combining two separate issues: the embedding in Euclidean space and the question of local trivialisation. It is preferable to think of $M$ as a $k$-dimensional differentiable manifold. Then the existence of a local chart means that one has a map $\psi$ between a neighborhood in $M$ and a neighborhood in $\mathbb R^k$. If $(x_1,\ldots,x_k)$ are the coordinates in $\mathbb R^k$, then $(\frac{\partial}{\partial x_1}, \ldots,\frac{\partial}{\partial x_k})$ is a basis for the tangent space of $\mathbb R^k$ at a point, and their image under $d\phi$ (or its inverse) will give a basis for the tangent space to $M$ at the corresponding point.