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 $n\_k$). 



Do partitions of unity exist if we impose additional conditions on the derivatives?
I hope the following construction will give you what you really need. If not, you'll have to explain why.


Take any nice locally finite covering $\mathbb R\subset\cup\_j U\_j$ and take any smooth partition of unity $1=\sum\_j\psi\_j^2$ subordinated to this covering. Take any smooth positive function $F$ on $\mathbb R$. Choose the numbers $n\_j>0$ so that $\sum\_j n\_j^2(\psi\_j^2+(\psi\_j')^2)\le F/2$. Define the smooth function $\theta$ by
$$
(\theta^{\,\prime})^2\sum\_j n\_j^2\psi\_j^2=F-\sum\_j n\_j^2(\psi\_j^2+(\psi\_j')^2)
$$  

Finally, associate with each $U\_j$ *two* functions $\varphi\_{j,0}=\psi\_j\cos\theta$ and $\varphi\_{j,1}=\psi\_j\sin\theta$ and enjoy the identities
$$
\sum\_{j,k}\varphi\_{j,k}^2=1, \quad \sum\_{j,k}n\_j^2(\varphi\_{j,k}^2+(\varphi\_{j,k}')^2)=F\,.
$$


