I have consulted the book Introduction to Partial Differential Equations by Folland (1995, Second Edition, Princeton University Press) for the proof of a certain theorem, but I am having trouble verifying a minor detail.
Before I present what I am having trouble with, let me present the theorem and the proof as it is written by Folland on page 13 of the aforementioned book. (If $V$ is an open subset of $\mathbb{R}^n$, Folland writes $C_c^\infty(V)$ for the space of smooth functions on $\mathbf{R}^n$ whose support is compact and contained in $V$.)
(0.19) Theorem Let $K \subset \mathbb{R}^n$ be compact and let $V_1, \dotsc, V_N$ be bounded open sets such that $K \subset \bigcup_1^N V_j$. Then there exist functions $\zeta_1, \dotsc, \zeta_N$ with $\zeta_j \in C_c^\infty(V_j)$ such that $\sum_1^N \zeta_j = 1$ on $K$.
Proof Let $W_1, \dotsc, W_N$ be as in Lemma (0.18). [Comment: this means that the $W_j$ are open, cover $K$, and $\overline{W_j} \subset V_j$.] By Theorem (0.17), we can choose $\phi_j \in C_c^\infty(V_j)$ with $0 \le \phi_j \le 1$ and $\phi_j = 1$ on $\overline{W_j}$. Then $\Phi = \sum_1^N \phi_j \ge 1$ on $K$, so we can take $\zeta_j = \phi_j/\Phi$, with the understanding that $\zeta_j = 0$ wherever $\phi_j = 0$.
For my applications I am only interested in demonstrating that the $\zeta_j$ belong to $C_c^1(V_j)$, and I am having trouble verifying that they are continuously differentiable; everything else is fine. What is more, in my applications I do not assume that the $V_j$ are bounded, and I would appreciate an answer not relying upon their boundedness. It should be noted that the $\overline{W_j}$ may still be taken compact in this case. What follows is my progress.
Pick $j \in I$, and define $A_j = \{ x \in \mathbb{R}^n : \phi_j(x) > 0 \}$, an open set. Note that $\text{supp } \phi_j = \overline{A_j}$, since $\phi_j \ge 0$ by construction.
Firstly, $\phi_j = 0$ on the open set $\mathbf{R}^n \setminus \overline{A_j}$, hence the same is true of $\zeta_j$. We see that $\zeta_j$ is continuously differentiable on $\mathbf{R}^n \setminus \overline{A_j}$.
Secondly, one has $\phi_j > 0$ and $\Phi > 0$ on $A_j$, and both of these functions are continuously differentiable. Therefore, the same must be true of $\zeta_j$ on the open set $A_j$.
What remains is to prove that the partial derivatives of $\zeta_j$ exist and are continuous on the boundary $\partial A_j$, and this is where I am having trouble. However, I have demonstrated that $\zeta_j = 0$ on $\partial A_j$. For consider a point $x \in \partial A_j$, and pick a sequence $(x_m)_1^\infty$ in $\mathbb{R}^n \setminus A_j$ converging to $x$. Using continuity of $\phi_j$, one has $0 = \phi_j(x_m) \to \phi_j(x)$, and so $\phi_j(x) = 0$, thus $\zeta_j(x) = 0$.
Can anyone help me verify that the partial derivatives of $\zeta_j$ exist and are continuous on $\partial A_j$?
It is good that you have trouble proving that the given function is continuously differentiable. It is not even continuous, let alone continuously differentiable.
The proof in the book is not sound as written. Consider the simple case $N=1$. Lemma 0.18 gives the statement in this case, but let us see what the given proof produces. The lemma gives a function $\phi_1\in C_c^\infty(V_1)$ for which $0\leq\phi_1\leq1$ and $\phi_1|_K\equiv1$. Now $\Phi=\phi_1$, so we get the function $\zeta_1=\phi_1/\Phi=\chi_{A_1}$. This is the characteristic function of the set $A_1$, which is not even continuous at $\partial A_1$! This issue is not unique to $N=1$, but is clearest in this case.
But hope is not lost: the proof can be repaired by choosing the functions slightly differently.
To make the proof work, one needs to apply an additional cutoff to $\zeta_j$. Let $\Phi$ denote the same sum as in your proof. By continuity and compactness arguments there is an open set $W$ so that $K\subset W$ and $\min_{\overline W}\Phi=m>0$. By lemma 0.18 there is a function $\psi\in C_c^\infty(W)$ for which $0\leq\psi\leq1$ and $\psi|_{K}\equiv1$. Define $\psi_j=\psi\phi_j/\Phi$.
On the set $K$ these functions sum to one as desired: $\sum_{j=1}^N\psi_j(x)=1$ for all $x\in K$. It remains to show that they are smooth. In the set $W$ the function $1/\Phi$ is a bounded smooth function. (This is not true in the whole space, since $\Phi$ is compactly supported!) Each $\psi_j$ is compactly supported in $W$, and in this set $W$ the function $\psi_j=\psi\phi_j/\Phi$ is smooth as the product of three smooth functions. Outside the set $W$ we declare $\psi_j=0$; this does not effect smoothness since $\psi_j$ is compactly supported in the open set $W$.
The point is that division by $\Phi$ is problematic when $\Phi$ is or goes to zero. Therefore it is useful to look at a set in which $\Phi$ is strictly positive (so that division is no issue) but is larger than $K$ (so that the desired constant value is not affected). This is precisely the purpose of the set $W$.
As a side remark, assuming boundedness of the sets $V_j$ has no effect on this result: the version for any open sets follows from the version for bounded open sets. Since $K$ is compact, it is contained in some large open ball $B(0,R)$. Assuming you have proven the result for bounded sets, you can apply it for $\tilde V_j=V_j\cap B(0,R)$. The resulting functions have the desired properties for the original sets $V_j$.