Let $(V,\Vert\cdot\Vert_V)$ and $(W,\Vert\cdot\Vert_W)$ be normed vector spaces and let $f\colon U\to W$ be a function defined on an open set $U\subseteq V$. $f$ is said to be differentiable at $x\in U$ if and only if there exists $A\in B(V,W)$ such that
$$\lim_{\Vert h\Vert_V\to0}\frac{\Vert f(x+h)-f(x)-A(h)\Vert_W}{\Vert h\Vert_V}=0\tag1$$
If $f$ is differentiable at $x$, $A$ is called the derivative of $f$ at $x$. It may be denoted by $Df(x)$.
Now, I understand this definition. My problem is, it is not equivalent to the "classical" one when dealing with real functions of one real variable, that is
$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}h\tag2$$
In this case, the derivative is a real number, not a linear map. The linear map $A$ is the function $t\mapsto f'(x)t$.
So is the derivative of $f\colon\Bbb R\to\Bbb R$ defined by $(1)$ or by $(2)$? Why is there this discrepancy between the usual definition and its "generalization" (it's not a generalization, since they're not equivalent in the specific case?