Math 205 Lecture 10 - Derivatives, their properties and effects on graphs, and derivative formulas

Welcome to lecture 10 of Math 205 - Elements of Calculus.

In tonight's lecture, we start by talking about when a derivative would NOT exist. It is a limit after all, non-existence is possible. We show algebraically and graphically what this would look like.

We then move on to talking about multiple derivatives, in particular, the second derivative, which has many useful applications. We talk about how the second derivative tells us about concavity and inflections and what some interpretations of these features might be.

We then take a step back and look at what derivatives tell us. What does the original function (the zeroth derivative) tell us, what about the first, what about the second? What do these tell us and how can they interpreted and what would they look like on a graph? We explore all this and summarize in a table.

We then move on to another chapter where we begin our discussion on derivative formulas. Using limits to compute derivatives can be cumbersome, so we look at formulas for a few functions that will make our lives easier. We look at the power rule and the rules for differentiating logs and exponentials. We also look at the properties of a derivative--which are the same as the properties of a limit.

We will pick up next time with more formulas and rules (product rule, quotient rule, etc) so that we can take derivatives of more complicated functions.