Tropical cyclone (TC) potential intensity (PI) theory has been extensively used for future climate change assessments of TC activity. PI theory has a well known approximate form, consistent with a Carnot cycle interpretation of TC energetics, which relates PI to mean environmental conditions: the difference between surface and TC outflow temperatures and the air-sea enthalpy disequilibrium. The changes in these conditions (the increase in air-sea disequilibrium, in particular) provide a physical reason for the robust increase in tropical-mean PI simulated in future climate projections. Quantitative assessments of future changes, in contrast, make use of a numerical algorithm based on the relationship between PI and convective available potential energy (CAPE). Here, a recently developed analytic theory for CAPE is used to present an alternative derivation of an approximate form of PI. The derivation offers insight into the limited sensitivity of PI to the atmospheric stratification in the free troposphere. The resulting CAPE-based approximate formula nearly recovers the previous approximate PI formula, and the new formula helps account for the weaker-than-expected sensitivity of PI to surface relative humidity changes. The new analytic CAPE-based PI builds confidence in previous numerical CAPE-based PI calculations that use climate model projections of the future tropical environment.