A Note on Karl Marx’s Mathematical Manuscripts The Recursive Determination of Higher-Order Derivatives of a Function
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Published
May 13, 2026
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Abstract
This paper highlights that Marx’s algebraic, infinitesimal-free and limit-free method of differentiation also involves an algorithm for the recursive determination of higher-order derivatives based on the Taylor– Lagrange Theorem. This fact is relevant to the ongoing debates both on the evaluation of Marx’s Mathematical Manuscripts and on alternative ways of approaching and teaching the history and subject matter of calculus
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Keywords
Marx’s preliminary and definitive derivatives, Methods of differentiation, Recursive determination of Marx’s higher-order derivatives, Taylor–Lagrange Theorem
References
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Glivenko, V. I. (1934) 2021. “The Concept of Differential According to Marx and Hadamard.” In Karl Marx and Mathematics, edited by P. Baksi, 37–45. London: Routledge.
Gou, X., and C. Xin. 2026. “Holes: Interpretation and Simplification of Concepts in the Teaching of Calculus.” International Journal of Mathematical Education in Science and Technology 57, no. 2, 384–392.
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Paliogiannis, F. C. 2025. “On the Remainder in Taylor’s Theorem.” The College Mathematics Journal 56, no. 3: 225–230.
Peano, M. G. 1889. “Une Nouvelle Forme du Reste dans la Formule de Taylor.” Mathesis. Recueil Mathématique à l’Usage des Écoles Spéciales et des Établissements d’Instruction Moyenne 9: 182–183.
Range, R. M. 2015. What is Calculus? From Simple Algebra to Deep Analysis. Singapore: World Scientific Publishing.
Range, R. M. 2022. “Calculus: A New Approach for Schools that Starts with Simple Algebra.” European Mathematical Society Magazine 124: 42–48.
Shen, S. S. P., and Q. Lin. 2014. “DD Calculus”. Accessed December 18, 2025. https://doi.org/10.48550/arXiv.1404.0070
Struik, D. J. 1948. “Marx and Mathematics.” Science & Society 12, no. 1: 181–196.
Suzuki, J. 2005. “The Lost Calculus (1637–1670): Tangency and Optimization without Limits.” Mathematics Magazine 78, no. 5: 339–353.
Zhang, J., and Z. Tong. 2017. “The Third Generation of Calculus (Calculus without Limit Theory).” In Electronic Proceedings of the 22nd Asian Technology Conference in Mathematics, December 15–19, 2017. Chungli, Taiwan: Chung Yuan Christian University. Accessed December 18, 2025. https://atcm.mathandtech.org/EP2017/invited/4202017_21538.pdf
Glivenko, V. I. (1934) 2021. “The Concept of Differential According to Marx and Hadamard.” In Karl Marx and Mathematics, edited by P. Baksi, 37–45. London: Routledge.
Gou, X., and C. Xin. 2026. “Holes: Interpretation and Simplification of Concepts in the Teaching of Calculus.” International Journal of Mathematical Education in Science and Technology 57, no. 2, 384–392.
Grabiner, J. V. 1983. “The Changing Concept of Change: The Derivative from Fermat to Weierstrass.” Mathematics Magazine 56, no. 4: 195–206.
Landen, J. 1764. The Residual Analysis; A New Branch of the Algebraic Art, Οf very extensive Use, both in Pure Mathematics, and Natural Philosophy. Book I. London.
Lin, Q., and W.-C. Yang. 2010. “Making Teaching Calculus Accessible.” Electronic Journal of Mathematics and Technology 4, no. 3: 197–211.
Marx, K. 1994. Mathematical Manuscripts: Together with a Special Supplement. Edited by P. Baksi. Calcutta: Viswakos Parisad.
McAndrew, A. 2010. “An Elementary, Limit-Free Calculus for Polynomials.” The Mathematical Gazette 94, no. 529: 67–83.
Okishio, N. 1993. Essays on Political Economy: Collected Papers. Edited by M. Krüger and P. Flaschel. Frankfurt am Main: Peter Lang.
Okuno-Fujiwara, M., and K. Shell. 2009. “An Interview with Hirofumi Uzawa.” Macroeconomic Dynamics 13, no. 3: 390–420.
Paliogiannis, F. C. 2025. “On the Remainder in Taylor’s Theorem.” The College Mathematics Journal 56, no. 3: 225–230.
Peano, M. G. 1889. “Une Nouvelle Forme du Reste dans la Formule de Taylor.” Mathesis. Recueil Mathématique à l’Usage des Écoles Spéciales et des Établissements d’Instruction Moyenne 9: 182–183.
Range, R. M. 2015. What is Calculus? From Simple Algebra to Deep Analysis. Singapore: World Scientific Publishing.
Range, R. M. 2022. “Calculus: A New Approach for Schools that Starts with Simple Algebra.” European Mathematical Society Magazine 124: 42–48.
Shen, S. S. P., and Q. Lin. 2014. “DD Calculus”. Accessed December 18, 2025. https://doi.org/10.48550/arXiv.1404.0070
Struik, D. J. 1948. “Marx and Mathematics.” Science & Society 12, no. 1: 181–196.
Suzuki, J. 2005. “The Lost Calculus (1637–1670): Tangency and Optimization without Limits.” Mathematics Magazine 78, no. 5: 339–353.
Zhang, J., and Z. Tong. 2017. “The Third Generation of Calculus (Calculus without Limit Theory).” In Electronic Proceedings of the 22nd Asian Technology Conference in Mathematics, December 15–19, 2017. Chungli, Taiwan: Chung Yuan Christian University. Accessed December 18, 2025. https://atcm.mathandtech.org/EP2017/invited/4202017_21538.pdf
Issue
Section
Marxism Studies
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How to Cite
A Note on Karl Marx’s Mathematical Manuscripts: The Recursive Determination of Higher-Order Derivatives of a Function. (2026). World Marxist Review , 3(1), 1−8. https://doi.org/10.62834/11pdjm96