Principles of Mathematical Analysis (3rd Edition)
by Walter Rudin · McGraw-Hill
Chapter 5 gives the compressed canonical treatment of differentiation: derivative of a real function, mean value theorems, continuity of derivatives via Darboux, L'Hopital's rule, Taylor's theorem, and differentiation of vector-valued functions.
This link may earn us a small commission at no extra cost to you. Affiliate disclosure
More resources on Differentiation
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Real Analysis (MIT 18.100A)
Rigorous foundations of calculus: real numbers, convergence of sequences and series, continuity, differentiability, the Riemann integral, sequences and series of functions, uniform convergence and interchange of limits. Includes 25 lecture videos, notes, problem sets and exams, and teaches how to read and construct proofs.
Continuous Nowhere Differentiable Functions
A 100-page survey tracing Bolzano's, Cellerier's, Weierstrass's and van der Waerden's pathological functions, with full construction proofs, and a Baire category argument showing such functions are generic among continuous functions. Directly attacks the hardest conceptual point in this topic - that differentiability is strictly stronger than continuity - by working the actual constructions instead of just naming Weierstrass.
Understanding Analysis (2nd Edition)
Chapter 5 develops the derivative motivated by the question of what functions can look like, proving the mean value theorem and Darboux's theorem, and showing why differentiability is strictly stronger than continuity.
Differentiable Functions (Chapter 8, An Introduction to Real Analysis)
A self-contained 30-page chapter covering the derivative's limit definition, chain rule, extreme values, the mean value theorem, Taylor's theorem, the inverse function theorem and L'Hopital's rule, with full proofs and worked counterexamples.
Introduction to Complex Analysis
Wesleyan University course on complex analysis covering complex numbers, analytic functions, conformal mappings, Möbius transformations, contour integration, Cauchy's theorem, power and Laurent series, and residues. Learners finish able to evaluate contour integrals and analyze complex-differentiable functions.