First pressure derivative of bulk modulus at zero pressure

[sirwan]

what is the physical meaning of first pressure derivative of bulk modulus at zero pressure. if it is ∂k/∂p, i say how it change with pressure, if it is at zero pressure?

 

[DrDu]

It means $\partial k/\partial p|_{p=0}=\im_{\Delta p \to 0}(k(0+\Delta p)-k(0))/\Delta p$
I.e. it is the change of bulk modulus when you increase the pressure slightly starting at p=0.
This is analogous to asking about the accelleration of a car dv/dt at zero speed.