MathTrackX: Differential Calculus

This module provides a foundation in the basic concepts and techniques of differential calculus. It covers the calculation of derivatives for polynomials and functions with real exponents, and explores practical applications of differentiation including finding slopes of curves, tangents, velocity, acceleration, and rates of change. Students will learn to identify key features of functions using derivatives and second derivatives, such as stationary points and inflection points, and will apply fundamental differentiation rules including sum, product, and quotient rules.

This module focuses on extending differentiation techniques to important non-polynomial functions used throughout mathematics and science. Students will learn how to calculate derivatives of standard trigonometric functions (such as sine, cosine, and tangent), exponential functions (including natural exponential and general exponential functions), and logarithmic functions. The module emphasizes applying standard differentiation rules to complex functions that incorporate these special functions, providing essential tools for modeling periodic phenomena, growth processes, and logarithmic relationships.

This module introduces students to the chain rule, a powerful technique for differentiating composite functions. Students will learn how to recognize when the chain rule is needed and how to apply it correctly to a variety of composite functions, including those involving polynomials, trigonometric functions, exponential functions, and logarithmic functions. The module emphasizes practical problem-solving using the chain rule in combination with previously learned differentiation techniques, enabling students to tackle more complex real-world applications that involve nested functions.

This module explores the real-world applications and advanced techniques of differentiation. Students will practice applying all differentiation rules learned throughout the course to solve complex problems involving products, quotients, and compositions of various function types. The module demonstrates how differential calculus can be used to derive rules for common special derivatives and solve practical problems in physics, engineering, economics, and other fields. Students will gain proficiency in selecting and applying the appropriate differentiation techniques based on the nature of the problem.

This module provides students with an opportunity to demonstrate their understanding of differential calculus through a timed examination. The assessment evaluates students' mastery of key concepts covered throughout the course, including basic derivatives, special function differentiation, the chain rule, and practical applications. Students will be tested on their ability to calculate derivatives using various techniques and apply differential calculus to solve problems. This summative assessment helps reinforce learning and validate students' comprehension of fundamental calculus principles.