It seems to me that two very different definitions of derivative permeate the literature: in introductory calculus, the derivative $f'(x)$ of a (differentiable) function $f:U\subseteq\mathbb{R}\rightarrow\mathbb{R}$ at a point $x\in\mathbb{R}$ is a real number. When one moves on to multivariable calculus, the derivative $Df(x)$ of a function $f:U\subseteq\mathbb{R}^m\rightarrow\mathbb{R}^n$ at a point $x\in\mathbb{R}^m$ is a linear mapping $Df(x):\mathbb{R}^m\rightarrow\mathbb{R}^n$.
However, when one talk about curves $\gamma:\mathbb{R}\rightarrow\mathbb{R}^n$, we are already in the domain of multivariable calculus, so the derivative $\gamma'(x)=D\gamma(x)$ of $\gamma$ at a point $x\in\mathbb{R}$ should be a linear mapping $D\gamma(x):\mathbb{R}\rightarrow\mathbb{R}^n$. But usually I see people using $\gamma'(x)$ as an element of $\mathbb{R}^n$. How to reconcile those?
My guess is that $\mathcal{L}(\mathbb{R},\mathbb{R}^n)\simeq M_{1\times n}(\mathbb{R})\simeq\mathbb{R}^n$, so people just identify $\gamma'(x)$ with its tuple in $\mathbb{R}^n$, just like sometimes one speaks of a matrix as a linear map or a linear map as being a matrix.
The relation between the velocity vector
$$ \gamma'(x) := \lim_{t \to 0} \frac{\gamma(x + t) - \gamma(x)}{t} $$
and the full differential $D\gamma|_{x} \colon \mathbb{R} \rightarrow \mathbb{R}^n$ is given by $\gamma'(x) = (D\gamma)|_{x}(1)$. Note that $D\gamma|_{x}(v)$ is the directional derivative of $\gamma$ at the point $x$ in the direction $v$, so it is defined by
$$ (D\gamma)|_{x}(v) := \lim_{t \to 0} \frac{\gamma(x + tv) - \gamma(x)}{t}. $$
By taking $v = 1 \in \mathbb{R}$, we get back the usual derivative. Since $D\gamma|_{x}$ is a linear map and $\mathbb{R}$ is one-dimensional, once you know the value of $D\gamma|_{x}(1)$, you know the value of $D\gamma|_{x}(c)$ for all $c \in \mathbb{R}$ using $$D\gamma|_{x}(c) = D\gamma|_{x}(c \cdot 1) = c D\gamma|_{x}(1).$$