Part II — Differentiation

"Nature is pleased with simplicity. And nature is no dummy." — Isaac Newton

In Part I we built the conceptual foundation. Now we compute.

The derivative is the first of the two fundamental operations of calculus. It measures how a function changes — its rate of change at every point. Geometrically, the derivative is the slope of the tangent line. Physically, the derivative of position is velocity, and the derivative of velocity is acceleration. Algorithmically, the derivative of a loss function tells a machine learning system how to improve. The derivative is everywhere because change is everywhere.

This part teaches you to compute derivatives — not as a mechanical skill, but as a fluency. By the end of Part II, you will be able to differentiate almost any function you can write down. You will use derivatives to find maxima and minima, sketch curves accurately, solve optimization problems from biology and economics, and approximate functions with their tangent lines.

What This Part Covers

  • Chapter 6 — The Derivative. Definition, notation, geometric meaning. Computing derivatives directly from the limit definition. When derivatives fail to exist. First appearance of the gradient-descent anchor.

  • Chapter 7 — Differentiation Rules. The rules that let you compute derivatives without going back to the limit definition. Power, sum, product, quotient, chain rules — derived, proved, and practiced extensively. Derivatives of trigonometric, exponential, and logarithmic functions.

  • Chapter 8 — Implicit Differentiation and Related Rates. When $y$ isn't isolated. When two quantities change simultaneously. The methodology: identify variables, write the equation, differentiate with respect to time.

  • Chapter 9 — Applications of Derivatives. Maxima and minima, the Mean Value Theorem, the first and second derivative tests, concavity, complete curve sketching, L'Hôpital's Rule.

  • Chapter 10 — Optimization. The most applied topic in Calc I. Setting up and solving optimization problems from physics, biology, economics, and engineering.

  • Chapter 11 — Linear Approximation and Newton's Method. The tangent line as the best local approximation. Newton's method for root-finding (the algorithm your calculator uses). Brief mention of $e^{i\pi}+1=0$ as a teaser.

  • Chapter 12 — Antiderivatives. The reverse problem: given a derivative, find the function. Antiderivative families. Indefinite integrals. The bridge to Part III: what is the connection between antiderivatives and areas?

What You Should Be Able to Do by the End of Part II

  • Compute the derivative of any elementary function by hand, fluently, in under 30 seconds
  • Use derivatives to find extrema, inflection points, asymptotes, and sketch accurate graphs
  • Solve real-world optimization problems from any field
  • Apply Newton's method to find roots numerically (and analyze its convergence)
  • Find antiderivatives of basic functions and recognize when a function has no elementary antiderivative
  • Use Python with sympy to verify hand differentiations and explore patterns

Why This Part Matters

Calculus has two engines: differentiation and integration. This part teaches the first. Everything in Calculus II and III builds on differentiation fluency. If you cannot compute derivatives quickly and accurately by the end of Part II, you will struggle for the rest of the book — not because integration is harder than differentiation, but because integration uses differentiation as a tool.

Spend the time. Do the exercises. The exercises in Chapter 7 in particular — there are many, and they look repetitive — are the single most important set of exercises in the book.

Computing derivatives should become as automatic as multiplying two-digit numbers. That fluency is the goal of Part II.

Chapters in This Part