 In this final video for math 1050 or lecture series here, I wanted to demonstrate how you can simplify a difference quotient that involves a radical, particularly a square root. This can get a little bit tricky, but so let's work through it here. If we start off with the average area of change, we get delta Y over delta X. What we see here is that we're gonna take F of X, let's say think of this as our function with a blank in it, right? G of blank, what does that mean? You mean you're gonna take the square root of blank squared plus a nine, okay? So that blank will sometimes be an X, sometimes we get an X plus H. So we're gonna get X plus H squared plus a nine, that all sits inside of a square root. And then we subtract from that a square root of X squared plus nine. This all sits above an H, like so. So how do you combine like terms on the top, when you have these difference of square roots and such? Well, as I've often mentioned in this lecture series, college algebra or pre-calculation, because whatever the name of the class is called, is often like the karate kid, right? We're Mr. Miyagi's teaching Daniels on all of these techniques. And so it's only until like the end that we actually discover like, oh, I was learning karate the whole time. So all these techniques we've learned in college algebra, intermediate algebra, elementary algebra, all of them come into play at some point. And so this one, it turns out we wanna rationalize the numerator. That is, I wanna multiply by the conjugate. So you have this square root minus the square root. Switch to sign the square root. I'm gonna take the square root of X plus H squared plus nine. I'm going to add to it, let me scooch over a little bit. I'm gonna add to it the square root of X squared plus nine, right? So it's the exact same difference of square roots. We've changed it to a sum. And you have to do the same thing to the denominator as well. So you get the square root of X plus H squared plus nine plus the square root of X squared plus nine. That's what you want there. So then in the denominator, you're gonna leave things factored. You never multiply out a denominator. It's not worth it, don't do it. So the denominator's gonna look like H times this big honkin thing. So the square root of X plus H squared plus nine plus the square root of X squared plus nine. You see that? Now in the numerator, I do want you to foil things out. So when we talk about foil, we have first, notice I'm gonna take the square root of X plus H squared plus nine times by itself. When you square a square root, the square root just disappears and you're left with X plus H squared plus nine. So that's the first thing. This is our first with our foil. Next, you're gonna multiply together the outside terms. When you multiply the outside terms, you get this one times that one, okay? Which I'm not gonna write that down because of the following reason. When you do the inside terms, you take this one times that one, you notice that the inside term and the outside term are actually identical to each other except they differ by a sign. You have the square root of X plus H squared plus nine times the square root of X squared plus nine. So basically the first one gives you the square root of A times the square root of B. The second one's gonna give you the negative square root of A times the square root of B. So you'll see that the outside and inside terms cancel each other out. That's why we multiply the conjugate. That's why we switched the sign. So I went from negative to positive because I knew when I foiled, these things would cancel out. And so then the last thing to do is the last terms, right? You take this one times that one, you get a square root of X squared plus nine times the square root of X squared plus nine, but you're squaring a square root that ends up with giving you a minus X squared plus nine. So we rationalize the numerator. All the square roots in the numerator disappeared. So we can actually simplify that thing a lot nicely. It's gonna be very nice, right? Take the X plus four squared, foil that out. We're going to get X squared plus two XH plus H squared plus nine. We then subtract from that the X squared plus nine and then copy down the denominator H times the square root of X plus H squared plus nine plus the square root of X squared plus nine. The denominator's just going for a ride right now. So in the numerator cancel like terms X squares cancel, the nine cancels like so. In which case then the numerator would simplify to be two XH plus H squared over H times that square root again, which I must write it down one more time. This is the hardest part about these difference quotients carrying around your luggage. The denominator, why do you have to do so complicated? All right, so the numerator now though, you'll notice everything, if you did everything right, everything should cancel out except for, well, after everything's canceled out, everyone in the numerator is divisible by H. So factor out the H because we need to cancel it out. You get H times two X plus H and then this is above H and then you get those square roots again, carry that heavy luggage up the stairs. I wish I didn't escalate it right now, right? I guess I already did that. Let's extend this line so that we see that there. And now the H cancels on the top and bottom for which then we're now ready to specify the simplified average rate of change, delta Y over delta X. This is then equal to, write out that denominator, make it a big, big, big line right there. You're gonna get two X plus H and then this sits above the square root of X plus H squared plus nine plus the square root of X squared plus nine, like so. This gives us our average rate of change. But then if we want the instantaneous rate change, right? We set H equal to zero. This is gonna give us a two X plus zero above the square root of X plus zero squared plus nine plus the square root of X squared plus nine. You notice the numerator simplifies just to be a two X but the denominator, you're gonna get the square root of X squared plus nine twice so you actually get two times the square root of X squared plus nine for which the twos actually cancel out and this thing simplifies kinda nicely, right? In the end, we end up with X over the square root of X squared plus nine, like so. And that then gives us this derivative, the so-called instantaneous rate of change. And that then brings us to the end of our lecture series. You can take out, you can look at some of the review videos we have for our final exam coming up as you try to prepare for that. That's a big deal. You wanna do well on that. Good at you for getting this far in our series here. And then again, I wanna congratulate you for making it to the end of the series. If you feel like during these many, many videos, if you learn something about college algebra, show that by hitting the like button. Feel free always to subscribe if you wanna see more videos like this in the future. Many of you watching this video, I know will be going on into a calculus class in the not too distant future at Southern Utah University. That's probably either math 1210, which is called calculus one, or it could be math 1100, which is a business calculus class. Or there's also an economics class that uses a lot of, I mean, there's a lot of economics, excuse me, a lot of calculus and economics. Some of them might be going in those directions. And so in the future, if you do need some help on calculus, feel free to check in with these videos in the future. I do have also a calculus one lecture series that you can take a look at. Those videos are available on YouTube. You can see the link for that right now. You can also get in the future if you continue on, depends how far you wanna go. But I also have a lecture series for, of course, calculus two, linear algebra, and a lot of other math classes. So feel free to subscribe if you wanna learn more about other math topics in the future if you're gonna continue in your math education. And always if there's questions, feel free to post them in the comments. And hopefully I'll see you all sometime in the future. Thank you, thank you for participating in this series and best of luck in your future mathematical endeavors.