Differentiation Summary

Real Analysis · Axiom Academy

How derivatives capture instantaneous change and enable rigorous analysis of function behavior — from the limit definition to the Mean Value Theorem and Taylor's theorem. Foundation in limits: the derivative is rigorously defined as a limit of difference quotients, making it a fundamental object in analysis. Differentiability is stronger than continuity: differentiable continuous, but not conversely (e.g. |x| at 0 ). Rules turn limits into algebra: linearity, product, quotient, and chain rules replace limit computations with mechanical manipulation. The MVT bridges local and global: it links a single point's derivative to behavior across a whole interval — the engine behind monotonicity, constancy, and inequalities. Taylor's theorem approximates: polynomials matching a function's derivatives, with a remainder term giving rigorous error bounds. The derivative measures the instantaneous rate of change at a point — geometrically, the slope of the tangent line, obtained as the limit of secant-line slopes (difference quotients) as the points coalesce. Key fact: differentiable at a continuous at a (the converse fails). Core Concept Differentiation Rules Linearity, the product, quotient, and chain rules convert limit computations into algebra. The power rule follows from these. Higher derivatives: differentiate repeatedly to get f''(x) , f'''(x) , Watch out for: the chain rule's inner factor g' — the most-forgotten piece.

This is the written version of the interactive lesson above. See the full Real Analysis course.