A Derivative Matrix is the Jacobian Matrix, and is Unique

Let $n$ and $m$ be natural numbers, let $\mathbb{R}$ be the real numbers, and let $U$ be an open subset of Euclidean space $\mathbb{R}^n$. Let $f:U\to\mathbb{R}^m$ with coordinate functions $f_1,\dots,f_m$ as in Jacobian Matrix of a Map Between Euclidean Spaces, let $a\in U$, and let $A$ be a real matrix with $m$ rows and $n$ columns, its entry in row $k$ and column $i$ written $A_{ki}$.

Suppose that $f$ is differentiable at $a$ with derivative matrix $A$. Then the following hold.

1. (Partial derivatives) For all natural numbers $k$ and $i$ with $1\le k\le m$ and $1\le i\le n$, the partial derivative of $f_k$ with respect to the $i$th variable exists at $a$ with value $A_{ki}$.

2. (Identification and uniqueness) The Jacobian matrix $Df(a)$ is defined, and $A=Df(a)$. In particular there is exactly one real matrix $A$ with $m$ rows and $n$ columns such that $f$ is differentiable at $a$ with derivative matrix $A$.