Calculus through Data & Modeling: Differentiation Rules

Johns Hopkins University via Coursera

Go to Course: https://www.coursera.org/learn/calculus-through-data-and-modelling-differentiation-rules

Introduction

**Course Review: Calculus through Data & Modeling: Differentiation Rules on Coursera** Mathematics is often termed the “language of the universe,” and calculus is a critical chapter in that language. If you’re looking to deepen your understanding of calculus, particularly differentiation, look no further than the Coursera course **"Calculus through Data & Modeling: Differentiation Rules."** This course is an essential continuation for anyone who has already grappled with the foundational concepts of calculus. ### Overview The course focuses on the foundations and applications of differentiation, a significant aspect of calculus concerned with rates of change. Specifically, it introduces new differentiation rules that allow students to compute derivatives efficiently without relying solely on the limit definition. This structured approach is ideal for those who want to strengthen their calculus skills and apply them to real-world problems, particularly in fields such as data science, economics, and engineering. ### Syllabus Breakdown 1. **Derivatives of Polynomial, Exponential, and Logarithmic Functions**: The course opens with an exploration of essential functions, including polynomials, exponentials, and logarithms. This module is crucial for setting the groundwork for understanding more complex derivatives and introduces differentiation rules designed to simplify calculations. 2. **The Product and Quotient Rules**: Students will learn how to differentiate functions created by the multiplication or division of other functions using these pivotal formulas. This section is ideal for paving the way for more advanced applications of derivatives in various contexts. 3. **Derivatives of Trigonometric Functions**: While reviewing trigonometric functions, students will delve into developing derivatives for sine and cosine. The course methodically builds out the catalog of derivative formulas to include all common trigonometric functions, essential for solving numerous practical problems involving rates of change. 4. **The Chain Rule**: This module introduces one of the most important concepts in differentiation: the chain rule. This rule enables students to find the derivatives of composed functions, a crucial tool for tackling real-world problems where multiple functions interact. 5. **Partial Derivatives**: Transitioning into multivariable functions, this module explores the concept of partial derivatives. This progression is significant for anyone interested in applying calculus to more complex data sets, and it introduces students to the geometric interpretations of these derivatives. 6. **Directional Derivatives and Gradient Vectors**: Building on the concepts of partial derivatives, students learn how to find rates of change in any direction through directional derivatives and gradient vectors. The applications of these concepts span various disciplines, including economics, biology, and physics, making this knowledge versatile and valuable. 7. **Final Project: Flight Path**: The course culminates in a hands-on project where students apply their knowledge to model a flight path for a landing aircraft. This practical application not only reinforces learning but also demonstrates how calculus can be used to solve real-world problems in aviation. ### Pros and Cons **Pros:** - **Structured Learning**: The course is well-organized, progressing naturally from single-variable functions to more complex multivariable calculus. - **Practical Applications**: Each module is intertwined with real-world applications, making the learning experience relevant and engaging. - **Project-Based Learning**: The final project encourages students to apply what they've learned, enhancing retention and understanding. - **Accessibility**: The course is available online, making it suitable for individuals with varying schedules. **Cons:** - **Pace of Learning**: Some students may find the pace challenging, especially those who are less familiar with foundational calculus concepts. - **Limited Interaction**: As with many online courses, lack of immediate feedback may hinder learners who benefit from direct interaction with instructors. ### Recommendation I highly recommend **"Calculus through Data & Modeling: Differentiation Rules"** for anyone seeking to consolidate their understanding of calculus, especially in the context of data and modeling. Whether you are a student, a professional looking to refresh your knowledge, or a data enthusiast, this course offers an accessible yet comprehensive approach to differentiable calculus. The blend of theoretical concepts and real-world applications ensures that you not only learn the material but can apply it effectively in practical scenarios. Dive into this course and equip yourself with powerful mathematical tools that will serve you well across various domains!

Syllabus

Derivatives of Polynomial, Exponential, and Logarithmic Functions

In previous course, we defined and calculated the derivative as a limit. In this module, we will examine the derivatives of some important functions, including polynomials, exponentials, logarithms, and trigonometric functions. We will also learn differentiation rules which will help us to compute derivatives more efficiently. Finally, we will generalize the idea of a derivative to multivariable functions, and learn how to find derivatives and rates of change on a graph in space.

The Product and Quotient Rules

The formulas of this section enable us to differentiate new functions formed from old functions by multiplication or division.

Derivatives of Trigonometric Functions

Before starting this module, please review trigonometric functions, in particular their graphs. In this module, we will develop formulas to find derivatives for the common trigonometric functions of sine and cosine. Together with the product and quotient rules, the derivatives for the remaining trigonometric functions are formulated. These new derivative formulas are then added to our catalog to use and apply to solve problems related to rates of change.

The Chain Rule

Many functions are created through composition of other functions. In this module, one of the most important of the differentiation rules of this course is developed which will allow us to find derivatives of the compositions of functions. This rule is called the chain rule and has a variety of applications.

Partial Derivatives

In this module, the notion of the derivative is applied to multivariable functions through the notion of partial derivatives. Algebraic rules are developed to find partial derivatives of multivariable functions as well as their geometric interpretations. The development of the tools of calculus to multivariable functions allows for further analysis of more complicated data sets.

Directional Derivatives and Gradient Vectors

In this module, we continue the application of partial derivatives to find rates of changes in any direction by developing the theory of directional derivatives and gradient vectors. These new tools of multivariable calculus can then be applied to problems in economics, physics, biology, and data science.

Final Project: Flight Path

Apply the theory of this course to model a flight path for a landing aircraft.