The derivative (or 'total derivative') a function $f : \mathbb{R}^m \rightarrow \mathbb{R}^n$ at some point $a \in \mathbb{R}^m$ is usually defined as a linear mapping $Df_a : \mathbb{R}^m \rightarrow \mathbb{R}^n$ which obeys $$ \lim_{h \to 0} \frac{f(a+h) - f(a) -Df_a(h)}{\Vert h \Vert_2} = 0. $$ This can be generalized to functions between Banach spaces. In the special case $m = n = 1$ this would mean that the derivative of $f$ at $a$ is a mapping from $\mathbb{R}$ to $\mathbb{R}$.
Now some authors (e.g. Munkres, Analysis on Manifolds) define the derivative as the $n \times m$ matrix $J_a(f)$ which obeys $$ \lim_{h \to 0} \frac{f(a+h) - f(a) -J_a(f) h}{\Vert h \Vert_2} = 0. $$
My question, therefore, is as follows:
What is the 'correct' definition of the derivative of $f$ at $a$? Is it the mapping $Df_a$ or the matrix $J_a(f)$?
You will agree that a linear map and its matrix (wrt some bases) are not the same thing. If $f : \mathbb{R} \rightarrow \mathbb{R}$, then the definition of the derivative as taught in elementary calculus and real analysis is Munkres' definition ($f'(a)$ is a $1 \times 1$ matrix). However if we have to use the first definition above, then we must define the derivative as the linear map $y \mapsto f'(a)y$.
Related question:
Both are correct and equivalent ( as long as we identify any linear map via its matrix-representation in base of standard vectors).
First one is more general, it can be used in any normed vector space, but you cannot use second one in infinite dimension vector space, since the size of a matrix can't be infinite.