What relates to multi-linearity in the same way differentiability relates to linearity? We know that a differential of a function existing at a point implies that said function can be approximated as a linear function near this point (local linearity).
What is the equivalent notion that allows a function to be approximated by a multilinear function locally , does such a notion even exist?
Side question that came to mind:
  Another way of looking a linearity is as a map that preserves a Vector space structure (maps a vector space into a vector space).
Analogously what abstract structure or space, tied with a structure preserving map, is related to multi-linearity.
Sorry if my Jargon is not accurate, but i hope the gist of my question is clear.  
 A: For later convenience, let $L^k(\mathbb{R}^n,\mathbb{R}^m)$ denote the vector space of all $k$-linear maps $(\mathbb{R}^n)^{k} \to \mathbb{R}^m$.
Let $f : U \to \mathbb{R}^m$ be a function defined on an open set $U \subseteq \mathbb{R}^n$. Recall that the derivative of $f$ at $\mathbf{x} \in U$ is the unique linear transformation $d_\mathbf{x}f : \mathbb{R}^n \to \mathbb{R}^m$, if it exists, such that
$$
 \lim_{h \to 0}\frac{\|f(\mathbf{x}+\mathbf{h})-f(\mathbf{x})-d_\mathbf{x}f(\mathbf{h})\|}{\|\mathbf{h}\|} = 0.
$$
If $f$ is actually differentiable on all of $U$, then the assignment $\mathbf{x} \mapsto d_\mathbf{x}f$ defines a function $f : U \to L^1(\mathbb{R}^n,\mathbb{R}^m)$, so that the second derivative of $f$ at $\mathbf{x} \in U$, if $d_\mathbf{x}(df) : \mathbb{R}^n \to L^1(\mathbb{R}^n,\mathbb{R}^m)$ exists, is the bilinear map $d^2_{\mathbf{x}} f \in L^2(\mathbb{R}^n,\mathbb{R}^m)$ defined by
$$
 \forall \mathbf{h}_1,\mathbf{h}_2 \in \mathbb{R}^n, \quad d^2_\mathbf{x}f(\mathbf{h}_1,\mathbf{h}_2) := \left(d_{\mathbf{x}}(df)(\mathbf{h}_2)\right)(\mathbf{h}_1).
$$
In general, by induction, if $f$ is $k-1$-times differentiable on all of $U$, then the $k$th derivative of $f$ at $\mathbf{x} \in U$, if $d_\mathbf{x}(d^{k-1}f) : \mathbb{R}^n \to L^{k-1}(\mathbb{R}^n,\mathbb{R}^m)$ exists, is the $k$-linear map $d^k_{\mathbf{x}} f \in L^k(\mathbb{R}^n,\mathbb{R}^m)$ defined by
$$
 \forall \mathbf{h}_1,\dotsc,\mathbf{h}_k \in \mathbb{R}^n, \quad d^k_\mathbf{x}f(\mathbf{h}_1,\mathbf{h}_2) := \left(d_{\mathbf{x}}(d^{k-1}f)(\mathbf{h}_k)\right)(\mathbf{h}_1,\dotsc,\mathbf{h}_{k-1}),
$$
and the appropriate version of Taylor's theorem now says that
$$
 f(\mathbf{x}+\mathbf{h}) = f(\mathbf{x}) + d_{\mathbf{x}}f(\mathbf{h}) + \frac{1}{2}d^2_\mathbf{x}(\mathbf{h},\mathbf{h}) + \cdots + \frac{1}{k!}d^k_{\mathbf{x}}f(\mathbf{h},\dotsc,\mathbf{h}) + o(\|\mathbf{h}\|^k), \quad \mathbf{h} \to 0.
$$
So now, decompose $\mathbb{R}^n$ as $\mathbb{R}^n = \mathbb{R}^{n_1} \times \cdots \times \mathbb{R}^{n_k}$ for some positive integers $n_1,\dotsc,n_k$ with $n_1 + \cdots n_k = n$. Suppose that $f : \mathbb{R}^{n_1} \times \cdots \times \mathbb{R}^{n_k} \to \mathbb{R}^m$ is actually $k$-linear. Then, in fact,
$$
 \forall \mathbf{x}, \mathbf{h} \in \mathbb{R}^n, \quad d^k_{\mathbf{x}}f(\mathbf{h},\dotsc,\mathbf{h}) = k! f(\mathbf{h}),
$$
whilst $d^lf = 0$ for $l > k$, so that $f$ is uniquely determined by its first $k$ derivatives at any point in $\mathbb{R}^n$. This suggests the following working definition for approximability by a $k$-linear map at a point:

Let $f : U \to \mathbb{R}^m$ be a $k$-times differentiable function on an open set $U \subseteq \mathbb{R}^{n_1} \times \cdots \times \mathbb{R}^{n_k}$. Let's define the $k$-linear derivative of $f$ at $\mathbf{x} \in U$ to be the unique $k$-linear map $[f]_{\mathbf{x},k} : \mathbb{R}^{n_1} \times \cdots \times \mathbb{R}^{n_k} \to \mathbb{R}^m$, if it exists, such that
  $$
  \forall 1 \leq j \leq k, \quad d^j_{\mathbf{0}}[f]_{\mathbf{x},k} = d^j_{\mathbf{x}}f.
$$

On the other hand, one can see, for instance, from Taylor's theorem, that two $k$-linear maps $g_1,g_2 :  \mathbb{R}^{n_1} \times \cdots \times \mathbb{R}^{n_k} \to \mathbb{R}^m$ are equal if and only if
$$
 g_1(\mathbf{h}) = g_2(\mathbf{h}) + o(\|\mathbf{h}\|^k), \quad \mathbf{h} \to 0,
$$
which suggests the following (nominal?) weakening of the first working definition:

Let $f : U \to \mathbb{R}^m$ be a function defined on an open set $U \subseteq \mathbb{R}^{n_1} \times \cdots \times \mathbb{R}^{n_k}$. Let's define the $k$-linear derivative of $f$ at $\mathbf{x} \in U$ to be the unique $k$-linear map $[f]_{\mathbf{x},k} : \mathbb{R}^{n_1} \times \cdots \times \mathbb{R}^{n_k} \to \mathbb{R}^m$, if it exists, such that
  $$
 \lim_{h \to 0}\frac{\|f(\mathbf{x}+\mathbf{h})-f(\mathbf{x})-[f]_{\mathbf{x},k}(\mathbf{h})\|}{\|\mathbf{h}\|^k} = 0.
$$

