The explicit formula for $\zeta(s)$: $$ \psi(x)=x-\sum_{|\operatorname{Im}\rho|<T}\frac{x^\rho}{\rho}-\log(2\pi)-\log\left(1-\frac{1}{x^2}\right)+O\left(\frac{x\log^2T}{T}\right), $$ where $\psi(x)=\sum_{p^k<x}\log p$, $x>1$, $\rho$ is a nontrivial zero of $\zeta(s)$, and the sum over $\rho$ is taken with multiplicities.
$\textbf{My question}:$ Is there any literature on the behavior of the equality when we take formal derivatives on both sides? I am looking for asymptotic results after taking the derivative on both sides.
(Expanding on my comment)
The classical way of recognizing this explicit formula is to perform a careful analysis of the integral $$ \psi(X) = \frac{1}{2\pi i} \int_{5 - i\infty}^{5 + i\infty} \left( - \frac{\zeta'(s)}{\zeta(s)} \right) X^s \frac{ds}{s}. $$ The convergence of this integral is a bit delicate, but it may be possible to define $\psi'(X)$ by defining it to be the value of the $X$-derivative of the integral.
A simpler problem would be to try to understand the integral $$ \frac{d}{dX} \frac{1}{2\pi i} \int_{5 - i\infty}^{5 + i\infty} X^s \frac{ds}{s}. $$ This behaves very well away from integer values of $X$, but at integer values this requires some interpretation.