If we differentiate the definition of the enthalpy ($H$), which depends on the internal energy ($U$), the pressure ($p$), and the volume ($V$), given by:

we obtain:

$dH = dU + Vdp + pdV$

using the differential enthalpy ($dH$), the internal energy differential ($dU$), the pressure Variation ($dp$), and the volume Variation ($dV$).

By differentiating the internal energy ($U$) with respect to the absolute temperature ($T$) and the entropy ($S$),

we get:

with the internal energy differential ($dU$) and the entropy variation ($dS$).

Therefore, it finally results in:

Given that the enthalpy ($H$) depends on the entropy ($S$) and the pressure ($p$), the differential enthalpy ($dH$) can be calculated using:

$dH = \left(\displaystyle\frac{\partial H}{\partial S}\right)_p dS + \left(\displaystyle\frac{\partial H}{\partial p}\right)_S dp$

To simplify this expression, we introduce the notation for the derivative of the enthalpy ($H$) with respect to the entropy ($S$) while keeping the pressure ($p$) constant as:

$DH_{S,p} \equiv \left(\displaystyle\frac{\partial H}{\partial S}\right)_p$

and for the derivative of the enthalpy ($H$) with respect to the pressure ($p$) while keeping the entropy ($S$) constant as:

$DH_{p,S} \equiv \left(\displaystyle\frac{\partial H}{\partial p}\right)_S$

thus we can write:

The differential enthalpy ($dH$) is a function of the variations of the entropy variation ($dS$) and the pressure Variation ($dp$), as well as the slopes the partial derivative of enthalpy with respect to entropy at constant pressure ($DH_{S,p}$) and the partial derivative of enthalpy with respect to pressure at constant entropy ($DH_{p,S}$), expressed as:

Comparing this with the equation for the differential enthalpy ($dH$):

we find that the slope of the partial derivative of enthalpy with respect to entropy at constant pressure ($DH_{S,p}$) with respect to the variation of the absolute temperature ($T$) is:

The differential enthalpy ($dH$) is a function of the variations of the entropy variation ($dS$) and the pressure Variation ($dp$), as well as the slopes the partial derivative of enthalpy with respect to entropy at constant pressure ($DH_{S,p}$) and the partial derivative of enthalpy with respect to pressure at constant entropy ($DH_{p,S}$), which is expressed as:

When compared to the equation for the differential enthalpy ($dH$):

it follows that the slope of the partial derivative of enthalpy with respect to pressure at constant entropy ($DH_{p,S}$) with respect to the variation of the volume ($V$) is:

Since the differential enthalpy ($dH$) is an exact differential, we should note that the enthalpy ($H$) with respect to the entropy ($S$) and the pressure ($p$) must be independent of the order in which the function is derived:

$D(DH_{S,p}){p,S}=D(DH{p,S})_{S,p}$

Using the relationship between the slope the partial derivative of enthalpy with respect to entropy at constant pressure ($DH_{S,p}$) and the absolute temperature ($T$)

and the relationship between the slope the partial derivative of enthalpy with respect to pressure at constant entropy ($DH_{p,S}$) and the volume ($V$)

we can conclude that: