Chapter 8 — Further Reading

Annotated pointers for going deeper on implicit differentiation and related rates. Start with the textbook mappings — they line up section-for-section with this chapter — then branch into the rigorous, applied, and historical readings.

Textbook Mapping (read these first)

  • Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.), Cengage.
  • §3.5 Implicit Differentiation — covers our §8.2–§8.5 (the method, second derivatives, inverse-trig derivatives). Stewart's worked curves (folium, cardioid, astroid) complement ours.
  • §3.6 Derivatives of Logarithmic Functions — includes logarithmic differentiation, our §8.6.
  • §3.9 Related Rates — the direct counterpart to our §8.7–§8.14; one of the richest related-rates problem sets in print (ladders, cones, searchlights, baseball diamonds). Work it for volume.

  • Full chapter-by-chapter correspondence in appendix-h-stewart-chapter-mapping.md.

  • Strang, G., and Herman, E. (2016). Calculus, Volume 1, OpenStax (free, openstax.org).

  • §3.8 Implicit Differentiation — our §8.2–§8.4, freely available with solved examples.
  • §3.9 Derivatives of Inverse Functions — our §8.5.
  • §4.1 Related Rates — our §8.7–§8.14, with the seven-step method laid out almost identically and many practice problems with answers.
  • Full correspondence in appendix-i-openstax-chapter-mapping.md.

Recommendation. If you have access to only one outside source, use the OpenStax §4.1 problem bank — it is free, answer-keyed, and pitched at exactly this chapter's level.

Rigor and Theory (for math majors)

  • Spivak, M. (2008). Calculus (4th ed.), Publish or Perish. The cleanest rigorous treatment of differentiation; read it for the careful version of why implicit differentiation is legitimate (it depends on the differentiability the Implicit Function Theorem guarantees).

  • Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.), McGraw-Hill. The Inverse Function Theorem and Implicit Function Theorem in full rigor (Chapter 9). This is the formal backing for the §8.4 Math Major Sidebar and the §8.5 inverse-derivative formula.

  • Spivak, M. (1965). Calculus on Manifolds, Benjamin. A short, demanding book where the Implicit and Inverse Function Theorems take center stage in $\mathbb{R}^n$ — the multivariable destination that Chapter 30 reaches.

Applications — Aviation & Tracking (Case Study 1)

  • Nolan, M. S. (2010). Fundamentals of Air Traffic Control (5th ed.), Cengage. The operational backdrop to the radar closure-rate problem; its separation-standards chapters show why $\dfrac{dD}{dt}$ matters in practice.

  • Kuchar, J. K., and Drumm, A. C. (2007). "The Traffic Alert and Collision Avoidance System," Lincoln Laboratory Journal 16(2). A readable engineering account of TCAS and its time-to-closest-approach ($\tau$) logic, built directly on the related rate we computed.

  • Blackman, S., and Popoli, R. (1999). Design and Analysis of Modern Tracking Systems, Artech House. The full noisy, multidimensional, Kalman-filtered version of the clean related-rates geometry — for readers who want to see how the textbook problem becomes a real system.

Applications — Medical Imaging (Case Study 2)

  • Eisenhauer, E. A., et al. (2009). "New response evaluation criteria in solid tumours: revised RECIST guideline (version 1.1)," European Journal of Cancer 45(2). The clinical standard for translating measured tumor dimensions into response assessment — the practical face of the $V = \tfrac43\pi r^3$ related rate.

  • Prokop, M., and Galanski, M. (2003). Spiral and Multislice Computed Tomography of the Body, Thieme. A radiology reference whose volumetry sections show the measurement pipeline behind the idealized sphere model.

Applications — Economics (worked in §8.15 and the exercises)

  • Varian, H. R. (2014). Intermediate Microeconomics (9th ed.), Norton. The standard intermediate text; its treatment of indifference curves, isoquants, and the marginal rate of (technical) substitution is implicit differentiation throughout, though it rarely names the calculus.

  • Simon, C. P., and Blume, L. (1994). Mathematics for Economists, Norton. Makes the calculus explicit — implicit differentiation, the Implicit Function Theorem, and comparative statics — for readers who want the economics and the mathematics in one place.

  • Stewart, J. Calculus, §3.9. Dozens of well-crafted problems, escalating in difficulty; the single best source for additional related-rates practice.

  • Larson, R., and Edwards, B. H. (2018). Calculus (11th ed.), Cengage. Strong engineering-oriented related-rates sets, with many tank, conveyor, and machine-part problems.

  • Spiegel, M. R., et al. Schaum's Outline of Calculus, McGraw-Hill. Inexpensive, solution-dense; good for sheer volume of worked related-rates examples when you want pattern recognition.

Historical Note

  • Descartes, R. (1637 / 1954). The Geometry (Dover reprint, trans. Smith & Latham). The folium of §8.2 first appears in Descartes's challenge to Fermat over tangent methods — a problem that, before the chain rule, was research-grade. Worth a look to appreciate how much implicit differentiation simplified.

A Practice Recommendation

The single most useful exercise is to do many related-rates problems — early attempts feel frustrating, but the seven-step pattern (§8.7) becomes routine with repetition. Two specific drills build fluency fast:

  1. Derive all six inverse-trig derivatives from scratch using implicit differentiation, watching the range-dictated sign each time (§8.5).
  2. Pick a moving system around you — a car approaching an intersection, a draining sink, your own shadow under a streetlamp — name the changing quantities, write their relating equation, and differentiate it with respect to $t$. This trains the modeling instinct the chapter is really about.