Stationary process and derivative

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

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

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

Thanks in advance for your help.