Proof of A Derivative Matrix is the Jacobian Matrix, and is Unique

Write $a=(a_1,\dots,a_n)$. For a real number $t$ and a natural number $i$ with $1\le i\le n$, let $u_i(t)$ be the point of $\mathbb{R}^n$ whose $i$th coordinate is $t$ and whose $l$th coordinate is $0$ for every natural number $l$ with $1\le l\le n$ and $l\ne i$. We use the properties of the Euclidean norm, the properties of the absolute value, the elementary order arithmetic of the ordered field $\mathbb{R}$, whose order is total by Ordered Field, and the properties of finite sums. For a real number $s$ we write $s^2$ for $s\cdot s$.

Step 1 (the norm of the coordinate increment). Let $t$ be a real number. By claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $\lVert u_i(t)\rVert$ is the unique nonnegative real number $r$ with $r^2=\sum_{l=1}^{n}\bigl(u_i(t)_l\bigr)^2$. In that sum every summand with $l\ne i$ equals $0\cdot 0=0$ and the $i$th summand is $t^2$, so the sum equals $t^2$ by claim 7 of Properties of Finite Sums. Moreover $|t|$ is nonnegative and equals $t$ or $-t$, both by claim 1 of Properties of the Absolute Value in an Ordered Field, and $(-t)\cdot(-t)=t\cdot t$, so $|t|^2=t^2$ in either case. Therefore

Step 2 (the increment point). By Sum of Points of $\mathbb{R}^n$ the sum of points of $\mathbb{R}^n$ is formed coordinatewise, so $a+u_i(t)$ is the point whose $i$th coordinate is $a_i+t$ and whose $l$th coordinate is $a_l$ for $l\ne i$. This is exactly the point $(a_1,\dots,a_{i-1},a_i+t,a_{i+1},\dots,a_n)$ occurring in Partial Derivative on a Euclidean Open Set. We write $a[t]=a+u_i(t)$.

Step 3 (the matrix acts on the coordinate increment). Let $k$ be a natural number with $1\le k\le m$. By Matrix-Vector Product,

Every summand with $l\ne i$ equals $A_{kl}\cdot 0=0$ and the $i$th summand is $A_{ki}\,t$, so the sum equals $A_{ki}\,t$ by claim 7 of Properties of Finite Sums.

Step 4 (claim 1). Fix natural numbers $k$ and $i$ with $1\le k\le m$ and $1\le i\le n$, and let $\varepsilon$ be a real number with $0<\varepsilon$. By claim 8 of Elementary Order Arithmetic in an Ordered Field the real number $\varepsilon'=\varepsilon\cdot 2^{-1}$ satisfies $0<\varepsilon'$ and $\varepsilon'<\varepsilon$. Applying Differentiability at a Point for Maps Between Euclidean Spaces with $\varepsilon'$ furnishes a real $\delta$ with $0<\delta$ such that every $h\in\mathbb{R}^n$ with $0<\lVert h\rVert<\delta$ satisfies $a+h\in U$ and $\lVert f(a+h)-f(a)-A\,h\rVert\le\varepsilon'\lVert h\rVert$.

Let $t$ be a real number with $0<|t|<\delta$ and put $h=u_i(t)$. By Step 1, $\lVert h\rVert=|t|$, so $0<\lVert h\rVert<\delta$. Hence $a[t]=a+h$ lies in $U$, and, writing $z=f(a[t])-f(a)-A\,h$,

Differences of points of $\mathbb{R}^m$ are formed coordinatewise, so the $k$th coordinate of $z$ is $z_k=f_k(a[t])-f_k(a)-(A\,h)_k$, which by Step 3 equals $f_k(a[t])-f_k(a)-A_{ki}\,t$. By claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ we have $|z_k|\le\lVert z\rVert$, so by claim 2 of Elementary Order Arithmetic in an Ordered Field,

Since $0<|t|$ we have $t\ne0$, so $t^{-1}$ exists. Put $w=z_k\cdot t^{-1}$. Then $z_k=w\cdot t$, so $|z_k|=|w|\,|t|$ by claim 4 of Properties of the Absolute Value in an Ordered Field, and therefore $|w|\,|t|\le\varepsilon'|t|$. We claim $|w|\le\varepsilon'$. Otherwise $\varepsilon'<|w|$, since the order is total, and then $\varepsilon'|t|<|w|\,|t|$ by claim 10 of Elementary Order Arithmetic in an Ordered Field, using $0<|t|$; combined with $|w|\,|t|\le\varepsilon'|t|$ this gives $|w|\,|t|<|w|\,|t|$ by claim 2 of that lemma, which is impossible. Hence $|w|\le\varepsilon'$, and $|w|<\varepsilon$ by claim 2 of that lemma together with $\varepsilon'<\varepsilon$.

Finally, distributing $t^{-1}$ over the three terms of $z_k$,

using $(A_{ki}\,t)\cdot t^{-1}=A_{ki}$. So every real $t$ with $0<|t|<\delta$ satisfies $a[t]\in U$ and

As $\varepsilon$ was an arbitrary positive real, this is precisely the condition of Partial Derivative on a Euclidean Open Set for the partial derivative of $f_k$ with respect to the $i$th variable to exist at $a$ with value $A_{ki}$. This proves claim 1.

Step 5 (claim 2). By claim 1 the partial derivative of $f_k$ with respect to the $i$th variable exists at $a$ for all $k$ and $i$ in the stated ranges, so the hypothesis of Jacobian Matrix of a Map Between Euclidean Spaces is met and the Jacobian matrix $Df(a)$ is defined. Its entry in row $k$ and column $i$ is $\partial f_k/\partial x_i(a)$, which equals $A_{ki}$ by claim 1 together with Uniqueness of the Partial Derivative on a Euclidean Open Set, the latter guaranteeing that the partial derivative has only the one value. Real matrices with $m$ rows and $n$ columns having equal entries throughout are equal, by Real Matrix and the Set of Real Matrices, so $A=Df(a)$.

For the uniqueness assertion, let $B$ be a real matrix with $m$ rows and $n$ columns such that $f$ is differentiable at $a$ with derivative matrix $B$. Applying what has just been proved to $B$ in place of $A$ gives $B=Df(a)$, and hence $B=A$.