>> Now, let's go through the process of finding the derivative function again. I want to show you the process from the top to the bottom. We have only done it one time and then after that we were talking about a single point, the slope of the tangent line at a particular point. Let's find the derivative function of this function. Now to find the derivative function or to find F prime of X we are using this form. This is the Delta X form, the limit of this difference quotient and also in using it like this, it is valuable to go out to the side and use the information about the function and evaluate this difference quotient. So that is what I am going to do. I am going to take the function, I am going to move it over here, and I am going to take our difference quotient and evaluate. Here is the quotient, here is the difference quotient, and here we go. And it is just a matter of using the idea of the function notation and filling in and performing the algebra that is involved. It turns out that it is the algebra in the calculus that makes calculus seem like it is difficult. If you are good at the algebra within the calculus, you are going to do just fine in calculus. Calculus is a series of concepts that are relatively easy to understand, but when you manipulate calculus, you are really just doing algebraic manipulations and they are just a little more intriguing or a little more challenging when you were taking college algebra or elementary algebra or whatever. It is just that the problems are a little longer, so you just have to have a little bit more room and write legibly and don't skip any steps. Okay, here we go. F of F plus delta X means in the F function replace all the Xs with this binomial, so it would be binomial squared minus two times binomial plus three, binomial squared minus two times binomial plus three. That is what this is all about. Alright then minus F of X means minus this, so it is minus, here it is. And now let's see squaring, it seems like we are squaring this binomial an awful lot, isn't it and we are. So X plus delta X all squared is X squared, plus 2 X Delta X plus Delta X squared and then minus 2 times X is minus 2X minus 2 times Delta X is minus 2 Delta X and then plus 3 and then minus all of this, all the signs change and we bring down the terms. Now we have a big string of terms here, but the nice thing about this is that every time you do it, if you do it 50 times, you will realize that you are looking for every term not involving Delta X to cancel, every term not involving Delta X. So X squared, there is bound to be a negative X squared somewhere, here it is. So that is basically what you are looking for, alright the minus 2X, we need a plus 2X, there it is and then this plus 3 minus 3, they go out. Alright, so all we have our terms that involve Delta X, now if all three terms involve Delta X, then we can factor the Delta X out and I am going to do that on the fly. I am going to do that between this step and the next step. We can see these terms clearly. Take Delta X out of this term and you are left with 2X. Take Delta X out of this term and you are left with the other Delta X factor. Take Delta X out of this term and you are left with minus 2. The Delta Xs cancel here, so we have this for the evaluation. Now we will bring this information back over here and continue with the problem, so F prime of X then is the limit as Delta X approaches 0 of, now this became what we have down there, the 2X plus Delta X minus 2 and as Delta X approaches 0 it goes out, so this then becomes 2X minus 2 and we can use this function to find the slope of the tangent line of any point that we would like to on the graph of that function.