For an ideal gas, you can derive it this way:
\begin{align}
\Delta S &= \int_{T_1,V_1}^{T_2,V_2}{\frac{\delta Q_\mathrm{rev}}{T}} \\
&= \int_{T_1}^{T_2}{\left(\frac{\delta Q_\mathrm{rev}}{T}\right)_V} + \int_{V_1}^{V_2}{\left( \frac{\delta Q_\mathrm{rev}}{T} \right)_T }\\
&= \int_{T_1}^{T_2}{\left(nC_V \frac{\mathrm{d}T}{T}\right)_V} +
\int_{V_1}^{V_2}{\left(\frac{p \cdot \mathrm{d}V}{T}\right)_T} &( \delta Q_\mathrm{rev})_T = (p.\mathrm{d}V)_T\\
&= nC_V \int_{T_1}^{T_2}{\left( \frac{\mathrm{d}T}{T}\right)_V} +
nR \cdot \int_{V_1}^{V_2}{\left(\frac{ \mathrm{d}V}{V}\right)_T} &(p=\frac{nRT}{V})\\
&= nC_V \ln {\left(\frac{T_2}{T_1}\right)} +
nR \ln {\left(\frac{V_2}{V_1}\right)} \\
\end{align}
If we consider an isobaric reversible thermal expansion instead, it would be:
\begin{align}
\Delta S &= nC_V \ln {\left(\frac{T_2}{T_1}\right)} +
nR \ln {\left(\frac{V_2}{V_1}\right)} \\
&= nC_V \ln {\left(\frac{T_2}{T_1}\right)} +
nR \ln {\left(\frac{T_2}{T_1}\right)}\\
&= n(C_V + R) \ln {\left(\frac{T_2}{T_1}\right)} \\
&= nC_p \ln {\left(\frac{T_2}{T_1}\right)}
\end{align}