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Calculus I: Limits & Derivatives

Where calculus starts: limits, continuity, and the derivative — built from the definition up through the product, quotient, and chain rules, then put to work on related rates, curve sketching, and optimization.

beginner Free calculusmathderivatives
📖 35 readings ⚡ 35 exercises · 485 min total
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4 modules · 35 lessons
0. Before Calculus 5 lessons · 68 min
  1. 1 The two problems that make calculus 13 min
  2. 2 Where πr² and 2πr come from 15 min
  3. 3 Functions, graphs, and the transformations that matter 14 min
  4. 4 Exponentials and logarithms 14 min
  5. 5 Trigonometry in radians 12 min
1. Limits and Continuity 10 lessons · 135 min
  1. 6 What a limit is — and isn't 13 min
  2. 7 Computing limits: the laws, factoring, and conjugates 13 min
  3. 8 When a limit fails: jumps, blow-ups, and oscillation 14 min
  4. 9 The squeeze theorem 12 min
  5. 10 The classic: sin(x)/x → 1 as x → 0 13 min
  6. 11 Its cousin: (1 − cos x)/x → 0 as x → 0 12 min
  7. 12 Limits at infinity, asymptotes, and growth rates 14 min
  8. 13 Continuity and its failure modes 13 min
  9. 14 The Intermediate and Extreme Value Theorems 15 min
  10. 15 ε–δ in practice 16 min
2. The Derivative 11 lessons · 154 min
  1. 16 The derivative as a limit 14 min
  2. 17 Differentiability vs. continuity 15 min
  3. 18 Power, constant-multiple, and sum rules 13 min
  4. 19 The product and quotient rules 14 min
  5. 20 The chain rule 13 min
  6. 21 Derivatives of the trigonometric functions 13 min
  7. 22 eˣ, ln x, and logarithmic differentiation 14 min
  8. 23 Inverse functions and the inverse trig derivatives 14 min
  9. 24 Hyperbolic functions, and why tanh shows up in neural nets 14 min
  10. 25 Implicit differentiation 14 min
  11. 26 Higher-order derivatives and what each one measures 16 min
3. Using the Derivative 9 lessons · 128 min
  1. 27 Related rates 13 min
  2. 28 Linear approximation and differentials 14 min
  3. 29 Newton's method 15 min
  4. 30 L'Hôpital's rule and the indeterminate forms 13 min
  5. 31 Rolle's theorem and the Mean Value Theorem 16 min
  6. 32 What f′ and f″ tell you 15 min
  7. 33 Curve sketching, end to end 13 min
  8. 34 Optimization: building the model, not just solving it 15 min
  9. 35 Numerical differentiation and the floating-point trap 14 min

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