This course is the 2nd course of my "Understanding Calculus" course series. At the end of this course, students will completely understand the following topics:
(1) Idea and formal definition of differentiation
(2) Derivative rules: product rule, quotient rule, chain rule, etc.
(3) Differentiability implies continuity.
(4) Derivative formulas of elementary functions, including polynomials, trigonometric functions, logarithmic functions, and exponential functions
(5) Important theorems related to differentiation: mean value theorem (MVT), 1st/2nd derivative tests, increasing/decreasing functions and their relation to differentiation, L'Hopital's rule
(6) Application to function sketching

This course is very mathematically rigorous in the sense that the proofs and their ideas are gone over and explained in details. Moreover, examples are also discussed, which build a concrete understanding of the topics that are introduced in the course. After completing this course, students will be able to confidently apply derivative rules and derivative formulas to compute the derivatives of functions that are composed of elementary functions. Also, students will be able to apply important theorems related to differentiation to compute limits in indeterminate forms using L'Hopital's rule.