 Hello, and welcome to the screencast where I'm going to show you how to use a spreadsheet to make numerical derivative estimates using the forwards, backwards, and central difference estimates. I've got an Excel spreadsheet open here with some data entered into it. However, everything you're about to see in this video can be done using Google spreadsheets, open office calc, numbers, or most other spreadsheets without any change in what you type. So, again, I have a function here called f of x, and it's not a formula or a graph, it's just a collection of seven data points. And notice that the x values happen to increase by five units each. So let's estimate f prime of 10, that's the value of the derivative of f at x equals 10. First of all, using a backwards difference. The derivative is estimated by an average rate of change between two points. For the backwards difference, I'm going to use as my two points, x equals 10 as one of the points, and then choose the closest x value that is behind that point as my second one, hence the name backwards difference. In this case, that's x equals five. So we're going to estimate this derivative for x equals 10, and to do that, I'm going to click on cell C4, which is in the row for x equals 10. Then since I'm going to have Excel calculate something, I'm going to first type an equal sign. And then type in the formula for the average rate of change from x equals five to x equals 10. As you remember, this is a fraction, and so I'm going to start with an open parenthesis for my numerator. What goes in the numerator is f of 10 minus f of five, and notice I'm clicking on the cells where f of 10 and f of five are located. Then I'm going to close the parenthesis, put in a division symbol, then open up another set of parenthesis for the denominator, and compute 10 minus five, again by clicking on the cells that hold 10 and hold five, and then close the parenthesis. Hitting enter at this point gives me the result, and that's the backwards difference approximation for f prime of 10. To get the forwards difference approximation for f prime of 10, I'll go click on cell D4 and have Excel calculate the average rate of change from x equals 10 to the next point forward from x equals 10 in the table, which is x equals 15. So click on that cell D4, type equals, open parenthesis, and then f of 15 minus f of 10. Again, I'm clicking on the cells containing those values. Close the parenthesis, put in the division symbol, open parenthesis for the denominator, and then 15 minus 10, then close the parenthesis and hit enter. And this is the forwards difference approximation for f prime of 10. Since we're in a situation where there's a data point in the table both in front of and behind x equals 10 at the same distance, namely x equals five and x equals 15, we can also use the central difference approximation for f prime of 10. To calculate this, I'm going to find the average rate of change from the point before x equals 10 to the point after x equals 10. So I'll go to cell E4, type equals, open parenthesis, f of 15 minus f of 5, close the parenthesis, put in the division symbol, open the parenthesis for the denominator, 15 minus 5, close parenthesis, and then hit enter. And that is the central difference approximation to f prime of 10. So I've cleared the screen off so I can show you why you might want to use a spreadsheet instead of a calculator to do this. Since spreadsheets allow us to apply formulas to large sets of data by dragging a formula through a range of cells, we can create a table of derivative estimates very quickly in this example. Let's start with the backwards difference, and suppose I wanted to calculate all of these entries in the backwards difference column all at once. Actually, I can only calculate the backwards differences for the final six cells. Since in the first cell where x equals zero, there's no data point behind x equals zero, and therefore a backwards difference is impossible. So I'm going to enter in the backward difference formula for x equals five, similarly to how we did it for x equals 10. And then when I'm done, I can just drag the formula through the remaining cells, and all the backwards differences are calculated. Similarly, I can calculate the forward difference for the first six cells in column D. In the last cell of column D, there is no data point ahead of x equals 30, and so I cannot calculate a forward difference. So I'll enter in the formula for the forward difference to estimate f prime of zero, then drag the formula through the remaining cells, and all those forward differences are calculated. Similarly, we can calculate the central difference for the cells in column E, except for the first and last ones. So that's how to use a spreadsheet to make tables of derivative estimates. Thanks for watching.