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.

5

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.
[FUTA, Akure] MTS 102: Introductory Mathematics II
[FUTA, Akure] MTS 102: Introductory Mathematics II
This learning track is structured for first-year students at the Federal University of technology, Akure (FUTA) and mirrors the standard second-semester coverage of elementary calculus. It begins with single-variable functions and their graphs, then walks learners through limits, continuity, differentiation techniques, and curve sketching—just as covered in the official MTS 102 outline. You’ll also explore anti-derivatives and integration, learning both the techniques and how to apply them to solve practical problems in science and engineering contexts. Everything is broken down into short, focused video lessons that keep things clear and manageable, especially for students who might be engaging this content for the first time. If you're not a FUTA student but need to build a solid foundation in these same topics, this track can serve you just as well. The structure and explanations are universal, ensuring that learners with similar academic goals can benefit fully.

This learning track is structured for first-year students at the Federal University of technology, Akure (FUTA) and mirrors the standard second-semester coverage of elementary calculus. It begins with single-variable functions and their graphs, then walks learners through limits, continuity, differentiation techniques, and curve sketching—just as covered in the official MTS 102 outline. You’ll also explore anti-derivatives and integration, learning both the techniques and how to apply them to solve practical problems in science and engineering contexts. Everything is broken down into short, focused video lessons that keep things clear and manageable, especially for students who might be engaging this content for the first time. If you're not a FUTA student but need to build a solid foundation in these same topics, this track can serve you just as well. The structure and explanations are universal, ensuring that learners with similar academic goals can benefit fully.

[OAU, Ife] MTH 201: Mathematical Methods I
[OAU, Ife] MTH 201: Mathematical Methods I
Comprehensive treatise of advanced calculus covering limits, continuity and differentiability, infinite sequences and series, partial differentiation, numerical methods and ordinary differential equations. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Comprehensive treatise of advanced calculus covering limits, continuity and differentiability, infinite sequences and series, partial differentiation, numerical methods and ordinary differential equations. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

[UNILAG, Akoka] MTH 102: Elementary Mathematics II
[UNILAG, Akoka] MTH 102: Elementary Mathematics II
This learning track is designed to guide first-year students at the University Of Lagos through key concepts in calculus, beginning with the fundamentals of single-variable functions and their graphs. It builds gradually into the core topics of limits, continuity, and differentiability, with each course tailored to simplify these foundational ideas for early learners. The focus is not just on theory but also on building the skill to solve problems confidently, especially those typically encountered in university-level exams. You’ll move from understanding the concept of a limit to mastering how derivatives work and how to apply them to sketch curves and analyze function behavior. Although built for UNILAG students, this track is suitable for anyone looking to strengthen their understanding of introductory calculus at the university level. Whether you're preparing for school assessments or seeking a solid refresher, this track will help you follow a structured path.

This learning track is designed to guide first-year students at the University Of Lagos through key concepts in calculus, beginning with the fundamentals of single-variable functions and their graphs. It builds gradually into the core topics of limits, continuity, and differentiability, with each course tailored to simplify these foundational ideas for early learners. The focus is not just on theory but also on building the skill to solve problems confidently, especially those typically encountered in university-level exams. You’ll move from understanding the concept of a limit to mastering how derivatives work and how to apply them to sketch curves and analyze function behavior. Although built for UNILAG students, this track is suitable for anyone looking to strengthen their understanding of introductory calculus at the university level. Whether you're preparing for school assessments or seeking a solid refresher, this track will help you follow a structured path.

Course Chapters

1. Introduction
2
2

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
2
2

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
3
1

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
2
3

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
2
5

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
3
3

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.