Calculus

  1. The Derivative
    1. Introduction to the Derivative
      1. The Tangent Line Problem
        1. Secant Lines
          1. Tangent Lines as Limits of Secants
          2. Rates of Change
            1. Average Rate of Change
              1. Instantaneous Rate of Change
              2. The Derivative as a Limit
                1. Definition Using Difference Quotients
                  1. Alternative Forms of the Definition
                  2. Interpreting the Derivative
                    1. Geometric Interpretation
                      1. Physical Interpretation
                        1. Rate of Change Interpretation
                      2. The Derivative as a Function
                        1. Derivative Notation
                          1. Leibniz Notation
                            1. Lagrange Notation
                              1. Newton Notation
                              2. The Graph of the Derivative
                                1. Relationship Between f and f'
                                  1. Sketching Derivative Graphs
                                  2. Differentiability
                                    1. Definition of Differentiability
                                      1. Relationship to Continuity
                                        1. Points of Non-Differentiability
                                          1. Corners
                                            1. Cusps
                                              1. Vertical Tangents
                                                1. Discontinuities
                                            2. Computing Derivatives
                                              1. Basic Differentiation Rules
                                                1. The Constant Rule
                                                  1. The Power Rule
                                                    1. The Constant Multiple Rule
                                                      1. The Sum and Difference Rules
                                                      2. The Product Rule
                                                        1. Statement and Proof
                                                          1. Extended Product Rule
                                                          2. The Quotient Rule
                                                            1. Statement and Proof
                                                              1. When to Use vs. Other Methods
                                                              2. The Chain Rule
                                                                1. Statement and Intuition
                                                                  1. Applications to Composite Functions
                                                                    1. Multiple Compositions
                                                                      1. Chain Rule in Different Notations
                                                                    2. Derivatives of Elementary Functions
                                                                      1. Derivatives of Polynomial Functions
                                                                        1. Power Rule Applications
                                                                          1. Higher-Order Polynomials
                                                                          2. Derivatives of Trigonometric Functions
                                                                            1. Derivative of Sine
                                                                              1. Derivative of Cosine
                                                                                1. Derivative of Tangent
                                                                                  1. Derivatives of Reciprocal Functions
                                                                                    1. Cosecant
                                                                                      1. Secant
                                                                                        1. Cotangent
                                                                                      2. Derivatives of Exponential Functions
                                                                                        1. Derivative of e^x
                                                                                          1. Derivative of a^x
                                                                                            1. Applications of Chain Rule
                                                                                            2. Derivatives of Logarithmic Functions
                                                                                              1. Derivative of ln(x)
                                                                                                1. Derivative of log_a(x)
                                                                                                  1. Logarithmic Chain Rule
                                                                                                  2. Derivatives of Inverse Trigonometric Functions
                                                                                                    1. Derivative of arcsin(x)
                                                                                                      1. Derivative of arccos(x)
                                                                                                        1. Derivative of arctan(x)
                                                                                                          1. Other Inverse Trigonometric Functions
                                                                                                        2. Advanced Differentiation Techniques
                                                                                                          1. Implicit Differentiation
                                                                                                            1. When to Use Implicit Differentiation
                                                                                                              1. Step-by-Step Process
                                                                                                                1. Higher-Order Implicit Derivatives
                                                                                                                2. Logarithmic Differentiation
                                                                                                                  1. When Logarithmic Differentiation is Useful
                                                                                                                    1. Products and Quotients
                                                                                                                      1. Variable Exponents
                                                                                                                      2. Derivatives of Inverse Functions
                                                                                                                        1. General Formula for Inverse Derivatives
                                                                                                                        2. Higher-Order Derivatives
                                                                                                                          1. Second Derivatives
                                                                                                                            1. Third and Higher Derivatives
                                                                                                                              1. Notation for Higher-Order Derivatives

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                                                                                                                          2. Limits and Continuity

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                                                                                                                          4. Applications of Differentiation