Introduction to Derivatives

An Introduction to Differential Calculus from High-School Mathematics

Overview

This introductory course treats differential calculus systematically, taking high-school mathematics as its starting point. Beginning from an intuitive understanding of rates of change, it progresses through differentiation techniques and on to applications in the analysis of functions. The later chapters — derivatives of inverse trigonometric functions, implicit differentiation, higher-order derivatives — reach beyond the standard high-school syllabus toward first-year university material.

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
  • 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, letting a variable approach a value, 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

    The sign of the derivative and an intuition for increase/decrease, shape of graphs, everyday applications (sign charts are treated systematically in Ch. 15)

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: Analysis of Functions and Applications (Chapters 15–18)

  1. Ch. 15 Monotonicity and Extrema

    Sign charts, local maxima and minima, the first and second derivative tests

  2. Ch. 16 Concavity and Inflection Points

    Determining concavity, inflection points, the second derivative and the shape of the graph

  3. Ch. 17 Optimization Problems

    Maximum and minimum problems, solution strategies

  4. Ch. 18 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)

Frequently Asked Questions

What is a derivative?

Differentiation is a way of studying how a function changes. The derivative at a point is the instantaneous rate of change there, which on a graph is the slope of the tangent line at that point. Collecting those values as a function of the point gives the derivative function. For example, the derivative of position with respect to time is velocity, and the derivative of velocity is acceleration.

What prerequisites are needed to study derivatives?

A knowledge of functions at the middle/high-school level, the ability to read graphs, and basic algebra are sufficient to begin. From Part 2 onward, a basic understanding of trigonometric, exponential, and logarithmic functions is also needed.

What does this introductory course cover?

Across 18 chapters, the course covers: an intuitive understanding of rates of change, the formal definition of the derivative, basic differentiation formulas, product/quotient/chain rules, derivatives of trigonometric/exponential/logarithmic functions, implicit differentiation, higher-order derivatives, analysis of monotonicity and concavity, and optimization problems.