 Hello, and welcome to the screencast about estimating the derivative. So given a table of values for a function, how can we estimate the value of the derivative at a certain point? Recall that you saw before that the forward and backward difference can be calculated with the formula, f prime of a is approximately equal to f of b minus f of a over b minus a. So the example we're gonna look at today is we're gonna estimate f prime of negative three using the different techniques. We're gonna do forwards, backwards, as well as central. And then I've given a table of x values here as well as some values of the function at those certain points. Okay, so we're gonna be focusing in on negative three. So to do the backwards difference, that means we're gonna wanna use the point before negative three as well as negative three. So then our backwards difference is gonna look like. So f prime at negative three is going to be approximately equal to f of negative three minus f of negative four. All over negative three minus a negative four. Okay, so that's just following the formula up here. Now I realize that the a value may not match with the a value here in the formula, but remember that forwards and backwards are just gonna be using the one before or after, just depending on which way you're doing. So this formula is just, like I said, kind of a rule to follow for that. Okay, so f of negative three by our table is four minus f of negative four by our table is 1.5. And then when you do the algebra and the denominator, that ends up giving you a value of positive one. So when you go ahead and simplify that, that gives you a value of 2.5. Okay, forward's difference. So we're now gonna use the value forward or in front of negative three. So in this case, that's gonna be negative two. So we're gonna do f of negative, oops, negative two minus f of negative three all over negative two minus negative three. And using the values off the table, f of negative two is 0.5 minus f of negative three is four, and that's all over a positive one. And that gives us a value of negative 3.5. Okay, so you notice the forward and backwards difference is quite different with this particular function. So let's take a look at the central difference. So central difference means you're gonna wanna think of negative three as the center. So f prime of negative three is about. That means you're gonna wanna use the values on both sides. And this time we're not gonna be using the value at negative three. And let's see if that gives us a little bit better estimate. So f of negative two minus f of negative four. Cuz those are gonna be the two values around the center of negative three. So negative two minus a negative four. My negatives would cooperate, there we go. And when I go and plug and chug those, that's gonna be 0.5 minus 1.5 all over two. So that gives us negative one over two, or negative 0.5. Okay, now what's interesting is the central difference can also be calculated given these two backwards and forwards differences. So if you notice, this is also equal to, I'll just kind of put this over here in brackets as an aside. If you were to take the average of the backwards and forwards, that would give you the central. So remember, average means you're gonna add those two values together. And if you divide by two, you'll also get negative 0.5. So I prefer obviously to do the calculation using the formula only because just in case if one of these two numbers were off, that means this number would also be off. So I prefer to use the raw data in the table. All right, thank you for watching.