Differentiability and Derivatives of Functions — Single-Variable Calculus

Here, you’ll learn what it means for a function to be differentiable and how to compute derivatives using first principles. We explore the derivative as a measure of instantaneous rate of change and as the slope of a tangent line. The course also highlights the relationship between differentiability and continuity, and includes geometric and physical interpretations of the derivative. This is where the core machinery of calculus really begins, with the derivative acting as a powerful tool for analysis.

21 hrs

$ 8.58

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

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

Chapter lessons

1.Slope of a line8:58

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

2.Differentiability at a point11:08

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

3.Worked examples (1)10:19

Some examples on the determination of the derivative of a function at a given point.

4.Worked examples (2)17:37

More examples on the determination of the derivative of a function at a given point.

2
Derivatives

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

Chapter lessons

1.Differentiability on an interval17:15

Derivative of a function over an interval.

2.Differentiability and its relation to continuity18:39

How continuity and differentiability are related.

3.Worked examples (1)53:13

Worked examples on the derivative of a function over an interval.

4.Worked examples (2)18:36

More worked examples on the derivative of a function over an interval.

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

1.Sums and products24:52

Derivatives of sums and products of functions.

2.Quotients25:58

Rule for differentiating the quotient of two functions.

3.Composites35:43

Rules for differentiating composite functions.

4.Worked examples (1)23:07

Worked examples on differentiation of functions.

4
Theorems on Differentiable Functions

Understanding the mean-value and Rolle's theorems.

Chapter lessons

1.Rolle's theorem57:35

The Rolle's theorem and its implications.

2.The mean-value theorem35:28

The mean-value theorem and its implications

3.Worked examples (1)28:29

Worked examples on the Rolle's and mean-value theorems.

4.Worked examples (2)29:53

Worked examples on the Rolle's and mean-value theorems.

5.Worked examples (3)25:43

Worked examples on the Rolle's and mean-value theorems.

5
Higher-Order Derivatives

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

Chapter lessons

1.Definition44:24

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

2.Worked examples (1)22:07

Worked examples on the evaluation of higher-order derivatives.

3.Worked examples (2)1:07:31

Worked examples on evaluation of higher-order derivatives.

4.Leibniz's formula1:18:57

Evaluating higher-order derivatives of a product of functions.

5.Worked examples (3)36:54

Worked examples on evaluation of higher-order derivatives using the Leibnitz's formula.

6.Worked examples (4)50:11

Worked examples on evaluation of higher-order derivatives using the Leibnitz's formula.

7.Worked examples (5)27:31

Worked examples on evaluation of higher-order derivatives using the Leibnitz's formula.

6
Taylor and Maclaurin Series

Taylor and Maclaurin series expansion of differentiable functions.

Chapter lessons

1.Maclaurin polynomials1:26:27

Polynomial approximations of differentiable functions.

2.Taylor polynomials35:41

Polynomial approximations of differentiable functions.

3.Taylor and Maclaurin series7:38

Infinite series representation of differentiable functions.

4.Worked examples (1)26:11

Worked examples on Taylor and Maclaurin series expansion of differentiable functions.

5.Worked examples (2)41:44

Worked examples on Taylor and Maclaurin series expansion of differentiable functions.

6.Worked examples (3)41:18

Worked examples on Taylor and Maclaurin series expansion of differentiable functions.