Differential Calculus · 73 lessons

A derivative is a limit, and taking that limit from first principles is slow. Each rule is the limit worked out once for one shape of expression. The skill is recognizing the shape: a power, a product, a quotient, or a function inside another function.

1. 1The Gradient of a Curve
2. 2Limits
3. 3From First Principles
4. 4Differentiating Powers
5. 5Tangents and Normals
6. 6Stationary Points
7. 7The Idea of a Limit
8. 8One Sided Limits
9. 9The Limit Laws
10. 10Indeterminate Forms
11. 11Limits at Infinity
12. 12The Squeeze Theorem
13. 13The Limit of sin x over x
14. 14Small-Angle Approximations
15. Limits (10)
16. 15Continuity at a Point
17. 16Types of Discontinuity
18. 17The Intermediate Value Theorem
19. Continuity (10)
20. 18The Derivative As a Limit
21. 19Derivative Notation
22. 20Differentiating From First Principles
23. 21Where a Derivative Fails to Exist
24. 22Differentiating Linear Functions
25. 23The Power Rule
26. 24The Constant Multiple and Sum Rules
27. 25The Product Rule
28. 26The Quotient Rule
29. 27The Chain Rule
30. 28Differentiating Trigonometric Functions
31. 29Differentiating tan x and the Reciprocal Ratios
32. 30The Number E
33. 31Differentiating Exponentials and Logarithms
34. 32Differentiating aˣ and logₐ x
35. 33Implicit Differentiation
36. 34The Derivative of an Inverse Function
37. 35Differentiating sin⁻¹ x and tan⁻¹ x
38. 36Higher Derivatives
39. Rules of Differentiation (10)
40. 37Tangents and Normals to Curves
41. 38The Tangent-Line Approximation
42. 39The Mean Value Theorem
43. 40Increasing and Decreasing Functions
44. 41Finding Stationary Points
45. 42Critical Points and Local Extrema
46. 43The Second Derivative Test
47. 44The First Derivative Test
48. 45Absolute Extrema on a Closed Interval
49. 46Points of Inflection
50. 47Curve Sketching with Derivatives
51. 48Optimization Problems
52. 49Motion in a Straight Line
53. 50Related Rates
54. 51Parametric Equations
55. 52Parametric Differentiation
56. 53Maclaurin Series
57. 54Where a Series Is Valid
58. 55L'Hôpital's Rule
59. 56From the Derivative Back to the Function
60. Using Differentiation (11)
61. 57Differentiating sinh x and cosh x
62. 58The Rate of Change of a Polar Function
63. 59Functions of Two Variables
64. 60Surfaces in Three Dimensions
65. 61Level Curves
66. 62Limits of Functions of Two Variables
67. 63Continuity in Two Variables
68. Functions of Several Variables (10)
69. 64The Partial Derivative
70. 65Computing Partial Derivatives
71. 66Higher Partial Derivatives
72. 67The Mixed Derivative Theorem
73. 68The Chain Rule for Partial Derivatives
74. 69The Gradient Vector
75. 70Directional Derivatives
76. 71Tangent Planes
77. 72Stationary Points of a Surface
78. 73Lagrange Multipliers