# Derivative

**Type:** Concept  
**Domain:** Analysis  
**Codex URL:** /mathematics/analysis/derivative/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Measures instantaneous rate of change; geometrically, the slope of the tangent line. Defined as a limit.

## Definition
f'(x) = lim(h→0) [f(x+h)−f(x)]/h

## Basic rules
xⁿ→nxⁿ⁻¹, sin(x)→cos(x), eˣ→eˣ, ln(x)→1/x
Sum, product, quotient, chain rules for combinations.

## Newton vs Leibniz
Newton developed calculus ~1665-66 (unpublished for decades). Leibniz developed independently ~1674-76, published first (1684). Bitter priority dispute followed. Modern consensus: independent discovery. Leibniz's notation (dy/dx) prevailed.

## Sources
### Tier 1
- Spivak, M. (2008). *Calculus*. 4th ed. Publish or Perish.
- Rudin, W. (1976). *Principles of Mathematical Analysis*. 3rd ed.
### Tier 2
- Hall, A.R. (1980). *Philosophers at War*. Cambridge University Press.
### Tier 3
- Boyer, C.B. (1949). *The History of the Calculus*. Dover.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*