Let $ ~~\cup_{k=-1}^{\infty} U_k = \mathbb{R} $ be an open covering of $\mathbb{R}$. It is a well known fact that partitions of unity subbordinate to the cover exists, i.e. there exists smooth functions $ \varphi_{k} : U_k \rightarrow \mathbb{R} $ with compact support such that $$ \sum_{k=-1}^{\infty} \varphi_k(x)^2 \equiv 1.$$

My first question is vague: Do there exist partitions of unity subbordinate to the cover if we impose some additional conditions on the derivatives (and the nature of the conditions are in terms of equalities, not inequalities.) ?

A more precise question is as follows:
Given a smooth function $f: \mathbb{R} \rightarrow \mathbb{R}$,
do there exist functions

$ \varphi_{k} : U_k \rightarrow \mathbb{R} $ with compact support
such that in addition to being a partition of unity subbordinate
to the cover, it also satisfies
$$ \sum_{k=-1}^{\infty} \varphi_{k}^{\prime}(x)^2 \equiv f(x)$$
? Here prime denotes derivative with respect to $x$.
I assume the answer should depend on what $f(x)$ is. Ideally
$\varphi_k$ should be smooth functions, but at the very least they ought
to be $C^1$.

I would prefer if we do not assume $f$ is nowhere vanishing, but if there is an answer assuming that, I would still like to see it.

Remark: The answer probably also depends on the open covering.

$\textbf{Very specific question:}$ Choose some number $\tau \in (\sqrt{2}, 2) $. Say $\tau = 1.5$. Now define $$ U_k = \{ x \in \mathbb{R}: \frac{2^k}{\tau} < |x| < 2^k \tau \} \qquad k=0,1,2, \ldots $$

$$ U_{-1} = (-1,1).$$

The collection $\{U_k\}_{k=-1}^{\infty} $ is an open covering of $\mathbb{R}$. I want smooth functions $\varphi_k: U_k \rightarrow \mathbb{R}$ and numbers $n_k$ such that $$ \sum_{k= -1}^{\infty} \varphi_k(x)^2 \equiv 1 $$ and $$ \sum_{k=-1}^{\infty} n_k^2 (\varphi_k(x)^2 + \varphi_k^{\prime}(x)^2) \equiv e^x$$

It is easy to see that if I set $n_k =1$ and $f(x) = e^x-1$, then it is the previous question I had asked. This question is "easier", because one is allowed to choose the numbers $n_k$ (in other words there is a better chance that the answer here might be yes, because of the freedom in choosing $n_k$).

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