In thermodynamics, an inexact differential is defined such that the integration of an inexact differential depends on the path between end points. [1] Differentials that do not fit this criteria are called exact diffferentials. Synonyms of inexact differential include “imperfect differential”, or “incomplete differential”, among others.

History
The subject of the exact and inexact differentials seems to be unique to the science of thermodynamics; introduced in the 1858 article “On the Treatment of Differential Equations which are not Directly Integrable” by German physicist Rudolf Clausius, later used an introductory section to the first (1865) and second (1875) editions of his thermodynamics textbook The Mechanical Theory of Heat. [2] The symbol đ (d-crossbar) or δ (in the modern sense) originated from the work of German mathematician Carl Neumann, specifically in his 1875 Lectures on the Mechanical Theory of Heat, indicating, as Clausius did, that δQ and δW are path dependent (inexact differentials), whereas internal energy dU is not (exact differential). [3]

References
1. (a) Potter, Merle C. and Scott, Elaine P. (2004). Thermal Sciences - an Introduction to Thermodynamics, Fluid Mechanics, and Heat Transfer, (pg. 67). U.S.: Brooks/Cole.
(b) Inexact Differential – from Wolfram MathWorld.
2. (a) Clausius, Rudolf. (1958). “On the Treatment of Differential Equations which are not Directly Integrable.” Dingler’s Polytechnisches Journal, vol. cl. (pg. 29).
(b) Clausius, Rudolf. (1879). The Mechanical Theory of Heat, (Section: Mathematical Introduction, pgs. 1-20). London: Macmillan & Co.
3. (a) Neumann, Carl. (1875). Lectures on the Mechanical Theory of Heat (Vorlesungen über die mechanische Theorie der Wärme), Germany.
(b) Laider, Keith, J. (1993). The World of Physical Chemistry (pg. 98). Oxford University Press.