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

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.

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.

MTH 207: Real Analysis I

Real Analysis bridges the critical gap between computational calculus and rigorous advanced mathematics. This learning track delivers the complete NUC CCMAS MTH 207 curriculum, transitioning you from intuitive understanding to formal mathematical proof. It establishes the theoretical foundation required for serious modelling in science, engineering, and pure mathematics. This programme is targeted at mathematics majors and advanced undergraduates in physics and engineering who have completed foundational calculus. It is designed for students requiring the rigorous analytical skills demanded by graduate-level studies and theoretical research. You will master the construction of rigorous proofs for sequence and series convergence, applying cornerstone theorems like Bolzano-Weierstrass and Cauchy criteria. You will achieve a formal command of continuity and differentiability, deriving major calculus rules from first principles. Completion provides the non-negotiable prerequisite knowledge for advanced studies in functional analysis, differential equations, and theoretical physics.

MTH 201: Mathematical Methods I

Mastering advanced calculus is essential for modelling complex systems in science and engineering. This track delivers the rigorous mathematical foundation demanded by the official NUC CCMAS curriculum for MTH 201. It systematically builds your expertise from fundamental single-variable theory to the sophisticated multivariable analysis used to solve critical problems in physics, economics, and technology. This programme is for undergraduates in engineering, mathematics, physics, and computer science requiring a deep theoretical and practical command of calculus. It also serves economics students needing advanced quantitative tools or professionals in finance and data science seeking a solid mathematical base for technical research. You will gain the analytical skills to construct formal proofs for differentiation rules and apply cornerstone theorems like Mean Value and Taylor's. You will master multivariable techniques, enabling you to solve constrained optimization problems with Lagrange multipliers and compute multiple integrals across line, surface, and volume domains. This track is the requisite preparation for advanced studies in differential equations, vector analysis, and complex engineering modelling.

Course Chapters

1. Introduction

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

Concept Overviews

2 Lessons

20:06

Problem Walkthroughs

2 Lessons

27:56

2. Derivatives

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

Concept Overviews

2 Lessons

35:54

Problem Walkthroughs

2 Lessons

1:11:49

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.

Concept Overviews

3 Lessons

1:26:33

Problem Walkthroughs

1 Lesson

23:07

4. Theorems on Differentiable Functions

Understanding the mean-value and Rolle's theorems.

Concept Overviews

2 Lessons

1:33:03

Problem Walkthroughs

3 Lessons

1:24:05

5. Higher-Order Derivatives

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

Concept Overviews

2 Lessons

2:03:21

Problem Walkthroughs

5 Lessons

3:24:14

6. Taylor and Maclaurin Series

Taylor and Maclaurin series expansion of differentiable functions.

Concept Overviews

3 Lessons

2:09:46

Problem Walkthroughs

3 Lessons

1:49:13