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?

Thanks in advance for your help.