 Welcome, let's quickly recap the main ideas from section 1.5 in active calculus on interpreting estimating and using the derivative. Previously we've defined the derivative of a function f of x to be the function f prime of x that's given by this limit formula. In this section we're mainly exploring what this function tells us and how it's related to its parent function f of x. First of all, we'll note that the units of f prime of x are the units of f of x divided by the units of x. And this is because the derivative is defined as a fraction, and the units simply follow the fraction. So for example, if f were measured in dollars and x is measured in time units like days, then f prime of x would be measured in dollars divided by days, which is better pronounced dollars per day. This helps us to understand what a derivative actually tells us in everyday terms. If f represents some function that depends on x, then f prime of x is telling us the rate at which f changes with respect to x. In other words, the rate at which f changes as x changes. This isn't any surprise because we originally thought of the derivative as an instantaneous velocity or as an instantaneous rate of change. Now the fact that f prime of x gives us the rate of change in f with respect to x itself has numerous applications, and we'll be working out those applications throughout much of the rest of this course starting with the activities in this section. The other main idea in this section is an extension of what we saw in the previous section. Back then we saw how to get f prime of x from f of x in two different ways. First of f is given as a formula, in which case we get f prime of x by setting up the derivative formula that uses a limit, working through the algebra and then taking a limit, and we also saw how to get a graph of f prime of x if the parent function f is given as a graph. The remaining representation we didn't talk about was what happens if f is given as a set of data. So we learned three ways in this section of finding f prime of a numerically at a given point x equals a, if f is given only as a table or collection of data points, and not as a graph and not as a formula. These are all estimation techniques and could include errors, but if we only have a table of data to work with, then estimation errors are simply unavoidable, so the best we can do is try to be as accurate in our estimations as possible. So we can estimate f prime of a using what's called a backwards difference or a forwards difference. Both of these differences involve using this definition of f prime of a, except without the limit. We can't take a limit if we can't choose two points to be arbitrarily close to each other, as is the case when f is a table of data. To estimate f prime of a, where a is some point for which the data is known, let a comma f of a be the known data point where you want the derivative, and then just choose b comma f of b to be some point that is ahead of a in the table. For the backwards difference, use b comma f of b or b as a point behind a in the table. Or if it's possible to look at symmetric data points equally spaced both before and behind a, we can use a central difference. To estimate f prime of a using a central difference, we go to a in the table and find a point that is ahead of it, and another point that is behind it at the same distance, and then calculate this fraction. In the next video, we're going to see an example of estimating a derivative value of a function using these estimation techniques.