Differentiability of Functions - Single-Variable Calculus (Undergraduate Foundation)

This course provides a rigorous, theoretical introduction to differential calculus. We will move beyond computation to understand the fundamental nature of the derivative, beginning with its formal definition as the limit of a difference quotient. The curriculum is designed to build a deep conceptual understanding of what it means for a function to be differentiable and the profound consequences of this property. The primary focus is on formal proofs and theoretical results. We will systematically derive the rules of differentiation, including the product, quotient, and chain rules, directly from first principles. This course emphasizes the "why" behind the mechanics of calculus, establishing the logical framework upon which all applications are built, including the cornerstone theorems of Rolle, the Mean Value Theorem, and Taylor's Theorem. By the end of this course, you will be able to: explain the derivative from its limit definition; prove the relationship between differentiability and continuity; formally derive all major differentiation rules; and understand the theoretical significance of the Mean Value Theorem and Taylor's Theorem. This course is designed for first-year undergraduates in mathematics, physics, and engineering who require a deep theoretical foundation for their studies. It is the ideal precursor to subsequent courses on differentiation techniques and applications, providing the essential logical underpinnings for those more practical subjects.

21 hrs

Payment required for enrolment

Enrolment valid for 12 months

This course is also part of the following learning tracks. You may join a track to gain comprehensive knowledge across related courses.

[OAU, Ife] MTH 201: Mathematical Methods I

This learning track delivers the complete mathematical toolkit required for a university-level science, engineering, or computing degree. It systematically covers the entire MTH 201 curriculum, building from the foundational principles of single-variable calculus - functions, limits, continuity, and differentiability - to the advanced methods of multivariable calculus, infinite series, numerical methods, and ordinary differential equations. This is the definitive preparation for advanced quantitative study. This programme is designed for second-year students offering MTH 201 at Obafemi Awolowo University, Ile-Ife, Nigeria. It is also helpful for any student in a STEM field - including physics, engineering, and computer science - who requires a rigorous and comprehensive command of calculus and its applications. This track delivers a full skill set in mathematical analysis and applied problem-solving. Graduates will be able to solve a wide range of problems, from optimising multivariable functions to modelling dynamic systems with differential equations and testing the convergence of infinite series. This programme directly prepares students for success in advanced courses in vector calculus, partial differential equations, and real analysis, providing the necessary foundation for a career in engineering, data science, or theoretical physics.

[NUC Core] GET 209: Engineering Mathematics I

Master the mathematical language of engineering. This programme delivers the complete analytical toolkit required for a successful engineering career, covering single-variable calculus, multivariable calculus, linear algebra, and vector analysis. It provides the essential foundation for all subsequent engineering courses. This programme is for second-year undergraduate students across all engineering disciplines. It delivers the official NUC CCMAS curriculum for Engineering Mathematics, providing the core training required for advanced modules in mechanics, thermodynamics, and circuit theory. Model and analyse complex physical systems using calculus, linear algebra, and vector analysis. You will be equipped to solve problems in dynamics, statics, and field theory, providing the quantitative proficiency required for advanced engineering study and professional practice.

[UI, Ibadan] MAT 223: Real Analysis

This learning track provides the complete theoretical machinery of single-variable calculus and analysis. We build the subject from first principles, establishing the rigorous logical framework required for advanced quantitative disciplines. This is the 'why' behind the mathematics that powers science and engineering. This track is built for second-year engineering and physical science students, particularly those at the University Of Ibadan. It is also structured for any student requiring the same rigorous theoretical foundation for advanced quantitative study. On completion, you will command the complete theoretical basis of single-variable calculus. You will construct formal proofs, rigorously analyse function behaviour, and determine the convergence of infinite series. This programme provides the non-negotiable prerequisite knowledge for advanced study in differential equations, complex analysis, and theoretical physics.

Course Chapters

1. Introduction

Meaning of the differentiability of real-valued functions at a point; illustration by slope of a straight line.

Chapter lessons

1-1. Slope of a line

8:58

A review of the meaning of the slope (gradient) of a straight line.

1-2. Differentiability at a point

11:08

Differentiability of a function, and its derivative at a given point.

2. Derivatives

Differentiability on an interval and the meaning of derivatives; relating differentiability and continuity of a function.

Chapter lessons

2-1. Differentiability on an interval

17:15

Derivative of a function over an interval.

2-2. Differentiability and its relation to continuity

18:39

How continuity and differentiability are related.

3. Rules of Differentiation

How to find the derivatives of real-valued functions; rules of derivatives of sums, products and quotients of functions and their proofs.

Chapter lessons

3-1. Sums and products

24:52

Derivatives of sums and products of functions.

3-2. Quotients

25:58

Rule for differentiating the quotient of two functions.

3-3. Composites

35:43

Rules for differentiating composite functions.

4. Theorems on Differentiable Functions

Understanding the mean-value and Rolle's theorems.

Chapter lessons

4-1. Rolle's theorem

57:35

The Rolle's theorem and its implications.

4-2. The mean-value theorem

35:28

The mean-value theorem and its implications

5. Higher-Order Derivatives

Higher-order derivatives of differentiable functions - meaning, proof of Leibniz's formula and its applications.

Chapter lessons

5-1. Definition

44:24

Meaning of higher-order derivatives and how to evaluate them.

5-2. Leibniz's formula

1:18:57

Evaluating higher-order derivatives of a product of functions.

6. Taylor and Maclaurin Series

Taylor and Maclaurin series expansion of differentiable functions.

Chapter lessons

6-1. Maclaurin polynomials

1:26:27

Polynomial approximations of differentiable functions.

6-2. Taylor polynomials

35:41

Polynomial approximations of differentiable functions.

6-3. Taylor and Maclaurin series

7:38

Infinite series representation of differentiable functions.