Introduction to Derivatives

Differential Calculus at the High-School Level

日本語版

Overview

This introductory course systematically covers the differential calculus taught in high-school mathematics. Starting from an intuitive understanding of rates of change, we progress through differentiation techniques and on to applications in the analysis of functions.

Learning Objectives

  • Gain an intuitive understanding of rates of change and limits
  • Understand the definitions of the derivative at a point and the derivative function
  • Master the basic differentiation formulas
  • Be able to use the product, quotient, and chain rules
  • Understand the mean value theorem and Taylor expansion
  • Be able to analyze monotonicity, extrema, and concavity of functions

Table of Contents

Part 1: Foundations of Differentiation (Chapters 1–6)

  1. Ch. 1 What Is Change?

    Velocity and distance, average rate of change, change on a graph

  2. Ch. 2 The Concept of Slope

    Slope of a line, secant and tangent lines, instantaneous slope on a curve

  3. Ch. 3 Intuition of Limits

    Intuitive understanding of limits, making things infinitely small, worked examples

  4. Ch. 4 Definition of the Derivative

    The derivative at a point, the derivative function, notation

  5. Ch. 5 Basic Derivatives

    Derivatives of constant, linear, and quadratic functions

  6. Ch. 6 The Meaning of the Derivative

    Determining increase/decrease, shape of graphs, everyday applications

Part 2: Differentiation Techniques (Chapters 7–14)

  1. Ch. 7 Product and Quotient Rules

    The product rule, the quotient rule, worked examples

  2. Ch. 8 The Chain Rule

    The chain rule, recognizing composite functions, practice problems

  3. Ch. 9 Derivatives of Trigonometric Functions

    Derivatives of sin, cos, tan; combining with the chain rule

  4. Ch. 10 Derivatives of Exponential Functions

    Derivative of $e^x$, derivative of $a^x$, the natural base

  5. Ch. 11 Derivatives of Logarithmic Functions

    Derivative of $\ln x$, derivative of $\log_a x$, logarithmic differentiation

  6. Ch. 12 Derivatives of Inverse Trig Functions

    Derivatives of $\arcsin$, $\arccos$, $\arctan$

  7. Ch. 13 Implicit Differentiation

    Implicit differentiation, parametric differentiation

  8. Ch. 14 Higher-Order Derivatives

    Second derivatives, higher-order derivatives, Leibniz's formula

Part 3: Theory and Applications (Chapters 15–22)

  1. Ch. 15 The Mean Value Theorem

    Rolle's theorem, the mean value theorem, Cauchy's mean value theorem

  2. Ch. 16 L'Hôpital's Rule

    Indeterminate forms, applying L'Hôpital's rule

  3. Ch. 17 Taylor Expansion

    Taylor's theorem, Maclaurin series, remainder terms

  4. Ch. 18 Expansions of Basic Functions

    Expansions of $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$

  5. Ch. 19 Monotonicity and Extrema

    Sign charts, local maxima and minima, the first derivative test

  6. Ch. 20 Concavity and Inflection Points

    Determining concavity, inflection points, the second derivative test

  7. Ch. 21 Optimization Problems

    Maximum and minimum problems, solution strategies

  8. Ch. 22 Approximation and Error

    Linear approximation, approximate calculations using derivatives, error estimation

Prerequisites

  • Knowledge of functions at the middle/high-school level
  • Ability to read graphs
  • Basic algebraic computation
  • Basics of trigonometric, exponential, and logarithmic functions (from Part 2 onward)