# 5.7 $\vartheta_b'[x]$: First-Order Derivative of First Chebyshev Function

The Fourier series for the first-order derivative $\vartheta_b'[x]$ of the first Chebyshev function evaluates to $2\ f\ \log[p]$ at primes of the form $x=p$ where $f$ is the evaluation frequency limit. The reference function in the following plots is $2\ f\ \log[Abs[x]]$.