The Definition of the Derivative

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Anything You Want to Talk About?

Are the grades for each outcome on problem sets independent of the grades for other outcomes? Generally, yes. I record separate grades for each outcome in the Canvas gradebook, and do the averaging of the highest two grades and the most recent one within each outcome. At the end of the semester, your course grade will just be a sum of those averages, weighted so that the maximum possible is 100%.

Misc

There is a new problem set available, on variations on limits (one-sided, infinite), and continuity.

This is nominally for this week, i.e., work on it towards the beginning of the week and grade it towards the end (or beginning of next week if your meetings with me are generally early in the week).

Defining the Derivative

From section 3.1 of the textbook and this discussion.

This definition captures the intuitive notion of a derivative as the slope of a tangent line in terms of limits:

This view focuses on the derivative as a number associated with a point on the graph of f(x). A slightly more powerful way to think of derivatives, which we’ll look at next, is to think of them as functions that can take any x value and return the derivative of f at that x:

Practice Using It

Using the limit definition of the derivative, try to find the derivatives of the following functions at the given places.

Notice that there are two formulations of the limit definition. Both say essentially the same thing, just in different ways. In particular, both calculate the slope of the secant line between two points of the graph of f(x), and ask what happens to that slope as the distance between the points goes to 0. In one formulation, the x coordinates of the points are considered separate things, a and x. In the other, the coordinate of the second point is calculated by adding a small amount to the coordinate of the first: a + h.

Find f′(0) where f(x) = x² + 2

We decided to use the limit as x approaches a version of the definition here. Finding the derivative then proceeded like so:

Find g′(2) where g(t) = 3/(2t)

Here we decided to try the limit as h goes to 0 version of the definition:

Whichever definition of the derivative you use, the process for using it to find a derivative f′(a) goes something like this:

1. (Write out the definition for reference.)
2. Plug x and a, or a+h and a, into the definition of f.
3. Use algebra to simplify the expression inside the limit until you eliminate division by 0.
4. Use limit laws to find the limit.

Next

There’s a better way to find derivatives at points: express the derivative as a function you can just plug x in to. And on the side, this also turns out to be a way to determine continuity that might be simpler than comparing a limit to a function’s value.

Read “Derivative Functions” and “Derivatives and Continuity” in section 3.2 of the textbook by class time Wednesday.

Also contribute to this discussion by class time Wednesday.

Wednesday will be a Cohort B day.

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