Proof of Chain Rule for $C^1$ Maps Between Euclidean Spaces

Fix a point $a\in U$. Since $f$ and $g$ are $C^1$ maps, C^1 Maps on Euclidean Open Sets are Differentiable shows that $f$ is differentiable at $a$ and that $g$ is differentiable at $f(a)$. Write

By Product of Real Matrices, the matrix product $BA$ is defined, and by Matrix-Vector Product the products $Ah$, $Bk$, and $Br(h)$ are defined for vectors of the appropriate sizes. We will prove that $g\circ f$ is differentiable at $a$ and that

Because $f$ is differentiable at $a$, for every $\eta>0$ there exists $\delta_1>0$ such that whenever $h=(h_1,\dots,h_n)\in\mathbb{R}^n$ satisfies $0<\sum_{i=1}^n h_i^2<\delta_1^2$ and $a+h\in U$, one has

Define the remainder vector $r(h)\in\mathbb{R}^m$ by

Then

Since the matrix $A$ has only finitely many entries, the real number

is well defined. For each $j\in\{1,\dots,m\}$,

Hence

Therefore, if we set

then

Now use differentiability of $g$ at $f(a)$. For every $\eta>0$ there exists $\delta_2>0$ such that whenever $k\in\mathbb{R}^m$ satisfies $0<\sum_{j=1}^m k_j^2<\delta_2^2$ and $f(a)+k\in V$, one has

Define the remainder vector $s(k)\in\mathbb{R}^p$ by

Then

Also set

For every vector $u=(u_1,\dots,u_m)\in\mathbb{R}^m$ we have

Choose $\eta>0$ so small that

Next choose $\delta_1$ and $\delta_2$ as above, and then choose $\delta>0$ so that

Let $h\in\mathbb{R}^n$ satisfy $0<\sum_{i=1}^n h_i^2<\delta^2$ and $a+h\in U$. Then $k=f(a+h)-f(a)$ satisfies $0<\sum_{j=1}^m k_j^2<\delta_2^2$, so the differentiability estimate for $g$ applies. Using $k=Ah+r(h)$, we obtain

Therefore,

Using the bounds above,

and

Hence

This is exactly the differentiability condition from Differentiability at a Point and Jacobian Matrix for Maps Between Euclidean Spaces. Thus $g\circ f$ is differentiable at $a$ with derivative matrix $BA$, that is,

Now let $x\in U$. Applying the first part with $a=x$, the $(\alpha,i)$ entry of the matrix identity gives

By the definition of C^1 Map on an Open Subset of Euclidean Space, each coordinate function $f_j$ is continuous on $U$. By Coordinatewise Characterization of Continuity for Euclidean Maps, the map $f$ is continuous on $U$ in the Euclidean sense. For each fixed $\alpha$ and $j$, the function

is continuous on $V$ by the definition of $C^1$. Hence Composition of Continuous Euclidean Maps shows that

is continuous on $U$. The function

is also continuous on $U$ by the definition of $C^1$. Therefore, by repeated use of Sums and Products of Continuous Real-Valued Functions, the right-hand side of the displayed formula is continuous on $U$. Thus every coordinate partial derivative of $g\circ f$ is continuous on $U$.

Finally, each coordinate function $g_\alpha$ is continuous on $V$ by the definition of $C^1$, so Composition of Continuous Euclidean Maps implies that each $g_\alpha\circ f$ is continuous on $U$. Hence, by the definition of C^1 Map on an Open Subset of Euclidean Space, the composition $g\circ f$ is of class $C^1$ on $U$.