So in the definition of bulk modulus the strain part is defined as (∆v/v) where ∆v is change in volume and v is the initial volume and while driving it for air we use the ideal gas law PV= constant and differentiate to to show that bulk modulus is equal to the initial pressure.
But in the gas law PV=k the "P" and "V" are instantaneous states, then why after differentiating do we set it equal to initial values and not the instantaneous ones? I understand that if ∆v is small then initial p is bulk modulus but what if ∆v large enough that the states change appreciably? Should the initial pressure still be considered the bulk modulus?
Note: I am assuming constant temperature.