Differentiation

Overview

Explore the fundamental theorems and proofs of differential calculus in this comprehensive 3.5-hour lecture series. Delve into rigorous mathematical proofs of essential concepts such as the product rule, quotient rule, and chain rule. Examine the critical Rolle's Theorem and Mean Value Theorem, understanding their implications and applications. Investigate the L'Hôpital's rule, including its proof and potential pitfalls. Analyze specific problems like the limit of sin(x)/x as x approaches 0, and discover the Banach Fixed Point Theorem. Gain insights into exponential properties and alternative problem-solving methods that don't rely on L'Hôpital's rule. Enhance your mathematical reasoning and deepen your understanding of calculus through these in-depth explorations of key differentiation concepts.

Syllabus

Proof of the product rule.
Quotient Rule Proof.
Proof of the Chain Rule.
Rolle’s Theorem Proof.
Mean Value Theorem Proof.
The Mean Value Theorem and Fixed Points.
Hopital rule proof.
lim sin(x)/x = 1 as x goes to 0.
Banach Fixed Point Theorem.
Nice L'Hôpital Problem.
Hopital Counterexample.
COOL Quotient Rule Proof.
Exponential Properties.
Don't use L’Hopital.