## Stationary process and derivative

Let $X_t,\ t\in \mathbb{R}$ a weakly stationary random process correlation function $b$ and spectral measure $\rho$.

Assuming $X_t(\omega)$ is differentiable and $X_t(\omega),\ X_t'(\omega)$ are bounded by a constant $c$ for almost all $\omega$, what is the correlation function and the spectral measure of $Z_t=X_t'$ ?

My main issue is with proving that $b$ is derivable. Do we need to show it is twice derivable to get the result?