--- title: Differentiation description: Differentiation is the study of how functions change locally, formalized through the concept of the derivative. Learners will understand the rigorous definition of differentiability, prove fundamental theorems of calculus, and analyze the local behavior of real-valued functions. category: mathematics subcategory: real-analysis difficulty: beginner, intermediate, advanced url: /subject/differentiation --- # Differentiation Differentiation is the study of how functions change locally, formalized through the concept of the derivative. Learners will understand the rigorous definition of differentiability, prove fundamental theorems of calculus, and analyze the local behavior of real-valued functions. ## Available Resources 2 Books • 2 Courses • 1 Websites • 2 Papers ## Courses ### 1. Real Analysis (MIT 18.100A) **Author:** Casey Rodriguez 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. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ **Tags:** real-analysis, proofs, sequences-and-series, uniform-convergence, riemann-integral ### 2. Introduction to Complex Analysis **Author:** Petra Bonfert-Taylor 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.coursera.org/learn/complex-analysis **Tags:** complex-analysis, analytic-functions, conformal-mapping, contour-integration, residue-theorem ## Papers ### 1. Continuous Nowhere Differentiable Functions **Author:** Johan Thim 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. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://ltu.diva-portal.org/smash/record.jsf?pid=diva2%3A1022983 **Tags:** real analysis, weierstrass function, nowhere differentiable functions, baire category theorem, pathological functions ### 2. Differentiable Functions (Chapter 8, An Introduction to Real Analysis) **Author:** John K. Hunter 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** http://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch8.pdf **Tags:** real analysis, chain rule, mean value theorem, taylor's theorem, inverse function theorem ## Books ### 1. Principles of Mathematical Analysis (3rd Edition) **Author:** Walter Rudin 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. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/007054235X?tag=edmonddante07-20 **Tags:** real analysis, mean value theorem, taylor's theorem, l'hopital's rule, rigorous calculus ### 2. Understanding Analysis (2nd Edition) **Author:** Stephen Abbott 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/1493927116?tag=edmonddante07-20 **Tags:** real analysis, mean value theorem, darboux's theorem, continuity, proof-based calculus ## Websites ### 1. Wolfram MathWorld **Author:** Eric W. Weisstein 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/differentiation