Chapter 4, Problem 4.40E

Equation $4.19$ says that

${\left(\frac{\partial U}{\partial V}\right)}_S=-p$

If we are considering the variation of $\Delta U$, the change in the change of the internal energy, we can write that as (see the previous problem for an analogous argument)

${\left(\frac{\partial \left(\Delta U\right)}{\partial V}\right)}_S=-\Delta p$

Show that this is entirely consistent with the first law of thermodynamics.