Dave4Math / Courses / Theory of Derivatives

Theory of Derivatives

Master the theory of derivatives in this AWESOME course on derivatives. In this course, you'll learn what derivatives are and become skilled at applying them.


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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Are you worried if you can do calculus? Have you ever thought about trying to understand it? Introducing Dave’s online math course: Theory of Derivatives. Dave helps you concentrate on both reading and writing calculus. Not only doing calculus (by solving problems) but also understanding it so that you can excel into the next level.

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in the theory of derivatives.
  • Students who are enrolled in calculus 1 and want to improve their grade.
  • Anyone interested in getting prepared for calculus 2.
  • Anyone wanting to learn about differentiation, the chain rule, implicit differentiation, related rates, and linear approximation.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is required.

What You’ll Learn in Theory of Derivatives

  • What is the Derivative?
  • Differentiation
  • Finding the Derivative of a Function
  • Using the Graph of f to Sketch the Graph of f’
  • Differentiability
  • Differentiability and Continuity
  • Some Basic Rules
  • The Derivative of the Natural Exponential
  • The Product Rule
  • The Quotient Rule
  • Extending the Power Rule
  • Higher-Order Derivatives
  • Motion Along a Line
  • Marginal Functions in Economics
  • Derivatives of Sines and Cosines
  • Derivatives of Other Trigonometric Functions
  • Composite Functions
  • The Chain Rule
  • Applying the Chain Rule
  • The General Power Rule
  • The Chain Rule and Trigonometric Functions
  • The Derivative of Exponentials
  • Implicit Functions
  • Implicit Differentiation
  • Derivatives of Inverse Functions
  • Derivatives of Inverse Trig Functions
  • Derivatives of Rational Powers of x
  • The Derivatives of Logarithmic Functions
  • Logarithmic Differentiation
  • The General Version of the Power Rule
  • The Number e as a Limit
  • Related Rates Problems
  • Solving Related Rates Problems
  • Increments
  • Differentials
  • Error Estimates
  • Linear Approximations
  • Error in Approximating

Course Description

We begin this course with the formal definition of the derivative of a function. Then, we give both the geometric interpretation and the physical interpretation of the derivative. Next, we discuss differentiation and its notation, and we work through several examples of finding the derivative of a function. After that, we examine the differentiability of a function and the relationship between differentiability and continuity.

In the next lesson, we talk about differentiation’s basic rules, including the linearity rule and the power rule. Then, we examine the derivative of the natural exponential function. After that, we study the product and quotient rules by working through several examples. We also discuss higher order derivatives and derivative notation.

We follow those lessons with the role of the derivative in the real world. In particular, we examine motion along the line, including velocity, speed, acceleration, and jerk. An introduction to marginal analysis and other applications are also included.

In the next lesson, we discuss the derivatives of the trigonometric functions followed by the derivatives of the composition of functions. The chain rule is covered in great detail, both its statement and proof, and we work through several examples of applying the chain rule. Derivatives of trigonometric functions, exponential functions, and logarithmic functions are highlighted.

An exciting part of the course is when we cover implicit differentiation and logarithmic differentiation. The derivative of an inverse function and derivatives of inverse trigonometric functions are covered in great detail. Towards the end of the course, we study related rates. In great detail, we discuss what makes a related rates problem and guidelines for solving a related rates problem; several examples are given in various categories. In the final lesson in this course, we investigate differentials and linear approximations.

Recommended Prerequisites for Theory of Derivatives

I recommend the prerequisite course Limits and Continuity. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free calculus 1 articles.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.