Here are 100 chapter titles for a book on Calculus Derivatives, progressing from beginner to advanced concepts, with a mathematical focus:
I. Foundations of Calculus & Limits:
- Introduction to Calculus: The Idea of Change
- Precalculus Review: Functions, Graphs, and Limits (Informal)
- The Concept of a Limit: Formal Definition
- Properties of Limits: Theorems and Examples
- Limits at Infinity and Infinite Limits
- Continuity: Definition and Properties
- Intermediate Value Theorem and its Applications
- Precise Definition of a Limit (Epsilon-Delta)
- Limits of Trigonometric Functions
- Indeterminate Forms and L'Hôpital's Rule
II. The Derivative: Basic Concepts:
- The Tangent Problem: Introduction to Derivatives
- The Derivative as a Limit: Formal Definition
- Interpretations of the Derivative: Slope, Rate of Change
- Differentiability and Continuity
- Rules of Differentiation: Power Rule, Constant Multiple Rule, Sum/Difference Rule
- Derivatives of Polynomial Functions
- Derivatives of Trigonometric Functions
- Derivatives of Exponential and Logarithmic Functions
- The Chain Rule: Composite Functions
- Implicit Differentiation
III. Techniques of Differentiation:
- Derivatives of Inverse Functions
- Derivatives of Hyperbolic Functions
- Derivatives of Inverse Trigonometric Functions
- Logarithmic Differentiation
- Derivatives of Parametric Equations
- Derivatives of Vector-Valued Functions
- Related Rates: Applications of Derivatives
- Linear Approximations and Differentials
- Higher-Order Derivatives
- Leibniz Notation and Higher-Order Derivatives
IV. Applications of Derivatives:
- Increasing and Decreasing Functions: The First Derivative Test
- Concavity and Inflection Points: The Second Derivative Test
- Optimization Problems: Finding Maximums and Minimums
- Curve Sketching: Using Derivatives to Graph Functions
- Mean Value Theorem and its Applications
- Rolle's Theorem
- Applications of Derivatives in Physics: Velocity, Acceleration
- Applications of Derivatives in Economics: Marginal Analysis
- Applications of Derivatives in Engineering
- Optimization in Multiple Variables (Introduction)
V. Limits and Derivatives: Deeper Dive:
- Limits Involving Exponential and Logarithmic Functions (Advanced)
- L'Hôpital's Rule: More Complex Cases
- Improper Integrals (Connection to Limits)
- Taylor Polynomials and Approximations
- Maclaurin Series
- Convergence and Divergence of Series
- Power Series and Taylor Series
- Applications of Series in Approximations
- The Binomial Series
- Fourier Series (Introduction)
VI. Advanced Differentiation Techniques:
- Implicit Differentiation: More Complex Examples
- Derivatives of Implicitly Defined Functions
- Partial Derivatives (Introduction)
- Directional Derivatives and Gradients
- The Chain Rule for Multivariable Functions
- Tangent Planes and Linear Approximations (Multivariable)
- Optimization in Multiple Variables (Advanced)
- Lagrange Multipliers
- Differentials and Error Estimation
- Implicit Function Theorem
VII. Applications of Derivatives (Advanced):
- Optimization Problems in Multivariable Calculus
- Related Rates in Multivariable Contexts
- Applications of Derivatives in Differential Equations (Introduction)
- Applications of Derivatives in Geometry: Curvature
- Applications of Derivatives in Physics: Vector Fields
- Applications of Derivatives in Economics: Optimization Models
- Applications of Derivatives in Computer Science
- Applications of Derivatives in Statistics
- Applications of Derivatives in Finance
- Applications of Derivatives in Biology
VIII. Differential Equations (Introduction):
- Introduction to Differential Equations: Basic Concepts
- First-Order Differential Equations: Separable Equations
- First-Order Differential Equations: Linear Equations
- First-Order Differential Equations: Exact Equations
- Applications of First-Order Differential Equations
- Second-Order Linear Homogeneous Differential Equations
- Second-Order Linear Nonhomogeneous Differential Equations
- Applications of Second-Order Differential Equations
- Series Solutions to Differential Equations
- Laplace Transforms (Introduction)
IX. Further Topics in Differential Calculus:
- The Mean Value Theorem: Extensions and Generalizations
- Taylor's Theorem: Remainder Estimation
- Convexity and Concavity (Advanced)
- Asymptotes and Curve Sketching (Advanced)
- Singular Points and Critical Points
- Optimization with Constraints
- Sensitivity Analysis
- Bifurcation Theory (Introduction)
- Chaos Theory (Introduction)
- Fractals and Dimension
X. Advanced Topics and Applications:
- Differential Geometry (Introduction)
- Calculus of Variations
- Optimal Control Theory
- Numerical Differentiation
- Symbolic Differentiation
- Applications of Calculus in Machine Learning
- Applications of Calculus in Image Processing
- Applications of Calculus in Signal Processing
- The History of Calculus
- The Future of Calculus and its Applications