5.1 $U_b'[x]=\delta[x+1]+\delta[x-1]$

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The Fourier series representation of $U_b'[x]$ (first-order derivative of $U[x]$) is illustrated below. Note $U_b'[x]$ corresponds to $\delta[x+1]+\delta[x-1]$ where $\delta[x]$ is the Dirac delta function. The first three plots illustrate the evaluation frequencies $f=1$, $f=2$, and $f=3$ respectively and include the left-half plane as well as the right-half plane. The fourth plot focuses around $x=1$ and includes multiple evaluation frequencies ($f\in\{2, 4, 8\}$). Note $U_b'[x]$ is an even function of $x$ (i.e. $U_b'[-x]=U_b'[x]$) and always evaluates to $2\ f$ at $x=1$ and to zero at other integer values of x.