Introduction to Limits
This MIT grad does a great job explaining limits and working examples. Good new, if you like this video the instructor has many more good calculus videos available.
Finding a Derivative Using the Definition of a Derivative
Patrick from JMT is a former Math Instructor at a top University. He is one of the most popular Math instructors on YouTube and this is one of his most popular calculus videos.
Calculus I in 20 Minutes (The Original) by Thinkwell
This video reviews the entire subject of Calculus in twenty minutes. The instructor is very enthusiastic and does a great job explaining this challenging subject.
What is calculus?
In this video Krista King gives her take on what calculus is. She describes herself as a calculus expert. She has many videos on calculus and is a gifted teacher. If you need help in calculus she would be a great resource.
Finding Real limits Graphical & Numerical Approach
This calculus video is created by Prof RobBob. He has a degree in Math and has taught calculus in high school for over 19 years. He is a gifted teacher and amazing handwriting. He always writes his entire lesson on a chalkboard. You will enjoy his videos.
What is Calculus? (Mathematics)
This video is produced by Socratica. I really enjoy all of their videos. This video starts with the question " Who invented calculus and goes from there.
In this list, I have identified the most popular calculus instructors on YouTube along with several great videos on the history and purpose of calculus.