{"id":"derivative","domain":"analysis","type":"Concept","title":"Derivative","slug":"derivative","url":"/mathematics/analysis/derivative/","summary":"Measures instantaneous rate of change of a function; geometrically, the slope of the tangent line. Defined as a limit. Half of calculus, invented independently by Newton and Leibniz.","formal_definition":"f'(x) = lim(h→0) [f(x+h)-f(x)]/h","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Spivak, M. (2008). Calculus. 4th ed. Publish or Perish."},{"tier":1,"citation":"Rudin, W. (1976). Principles of Mathematical Analysis. 3rd ed. McGraw-Hill."},{"tier":2,"citation":"Hall, A.R. (1980). Philosophers at War: The Quarrel between Newton and Leibniz. Cambridge University Press."},{"tier":3,"citation":"Boyer, C.B. (1949). The History of the Calculus and its Conceptual Development. Dover."}],"relationships":[{"type":"built_from","target":"limit","label":"Limit"},{"type":"inverse_of","target":"integral","label":"Integral — Fundamental Theorem of Calculus"},{"type":"applied_to","target":"polynomial","label":"Polynomial"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.4"}