Chapter 7 — Further Reading

An annotated guide to going deeper on the differentiation rules. Start with the textbook mappings to align this chapter with the two reference texts the book is built against (see _continuity.md §8); the rest is enrichment.

Mapping to the Standard Textbooks

Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.). Cengage. The reference text for breadth and exercise depth. This chapter corresponds to Stewart's §3.1–3.6: - §3.1 derivatives of polynomials and exponentials (our 7.2, 7.7) - §3.2 product and quotient rules (our 7.4, 7.5) - §3.3 derivatives of trigonometric functions (our 7.7) - §3.4 the chain rule (our 7.6) - §3.6 derivatives of logarithmic functions and logarithmic differentiation (our 7.7, 7.9)

Stewart is the gold standard for sheer volume of drill problems — worth the library copy for practice even though the new edition is expensive.

Strang, G., & Herman, E. (OpenStax, free). Calculus, Volume 1. A genuinely free, high-quality alternative. The matching material is in Chapter 3 (Derivatives): - §3.3 Differentiation Rules (power, product, quotient — our 7.2, 7.4, 7.5) - §3.5 Derivatives of Trigonometric Functions (our 7.7) - §3.6 The Chain Rule (our 7.6) - §3.9 Derivatives of Exponential and Logarithmic Functions (our 7.7, 7.9)

Every section has worked examples and free exercises with answers; download it before paying for anything.

Spivak, M. (2008). Calculus (4th ed.). Publish or Perish. For math majors who want the rules built rigorously. Chapters 9–10 develop differentiation with full proofs, including the airtight chain-rule argument sketched in the Math Major Sidebar of Section 7.6.

Apostol, T. M. (1967). Calculus, Volume I (2nd ed.). Wiley. An even more rigorous, theorem-first treatment (Chapter 4). Demanding but rewarding if you want analysis-grade care.

The Chain Rule, Visualized

3Blue1Brown (Grant Sanderson). "Visualizing the chain rule and product rule" — Essence of Calculus, Chapter 4. YouTube. The best intuition-builder available: animates the product rule as a growing rectangle and the chain rule as composed rates of change, exactly matching the Geometric Intuition callouts of this chapter.

Spivak, M. Calculus, Chapter 10. The careful derivation, for readers who found the §7.6 sidebar's auxiliary-function trick intriguing and want the full argument.

On Backpropagation (Case Study 1)

Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). "Learning representations by back-propagating errors." Nature, 323, 533–536. The classic paper that made backpropagation famous. Short, readable, and recognizably "just the chain rule."

Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning. MIT Press. Chapter 6. The standard graduate reference on feedforward networks and backprop; free online at deeplearningbook.org.

LeCun, Y., Bengio, Y., & Hinton, G. (2015). "Deep learning." Nature, 521, 436–444. A high-level review by three pioneers; good context for why the chain rule matters at scale.

Karpathy, A. "Building micrograd." YouTube. Implements an automatic-differentiation engine from scratch, one chain-rule step at a time — the clearest way to watch Case Study 1 actually run.

On Exponential Decay and Pharmacokinetics (Case Study 2)

Rowland, M., & Tozer, T. N. (2011). Clinical Pharmacokinetics and Pharmacodynamics (4th ed.). Lippincott Williams & Wilkins. The standard clinical text; its opening chapters build the same one-compartment $C(t) = C_0 e^{-kt}$ model used in Case Study 2, including half-life and multiple-dose accumulation.

Maor, E. (1994). e: The Story of a Number. Princeton University Press. The definitive popular history of $e$ — from compound interest to continuous decay — and why a function equal to its own derivative governs growth and decay everywhere.

Strogatz, S. (2019). Infinite Powers. Houghton Mifflin Harcourt. A general-audience tour of calculus; the chapters on exponential change explain the self-derivative of $e^x$ that drives every $e^{kt}$ model.

On Logarithmic Differentiation

This technique lives mostly in textbooks. Stewart §3.6 and OpenStax §3.9 both cover it with worked examples (including $x^x$); Apostol §6.6 gives a more theoretical treatment. It also reappears in financial mathematics under "log-returns."

On Automatic Differentiation (the Modern Chain Rule)

Baydin, A. G., Pearlmutter, B. A., Radul, A. A., & Siskind, J. M. (2018). "Automatic differentiation in machine learning: a survey." Journal of Machine Learning Research, 18, 1–43. The definitive survey of how the chain rule is implemented in software (forward vs. reverse mode).

Griewank, A., & Walther, A. (2008). Evaluating Derivatives (2nd ed.). SIAM. The book-length authority on computational differentiation.

JAX documentation. https://jax.readthedocs.io/ A leading modern AD library; its tutorials are an excellent way to see the chain rule of Section 7.6 executed at scale.

A Note on Practice

Chapter 7 is the most-practiced chapter in the book, and there is no shortcut — differentiation is a skill, and skills require repetition. Stewart offers the largest problem bank; Khan Academy (free) has well-curated, instantly-graded exercises on each rule; Paul's Online Math Notes (free) provides clean worked solutions. Spend the hours now. By Chapter 8, differentiation should be as automatic as arithmetic.