 In this video, I want to talk about how you can simplify difference quotients for which the function in play is itself a rational function. So there's some fractions involved in some kind. So if you want to evaluate this one, delta y over delta x, the difference quotient, you're going to be taking this to be f of, let's let me back up for a second here. So one thing to remember when you have these functions, think of it not as a variable, think of it as just a placeholder, right? So we're thinking of what is f of blank, right? This should look like one over blank squared. That's what we're going to do here. So in your numerator, you're going to have a one over blank squared minus one over blank squared. The denominator is always an h. And then what goes in the blank for the first one, you get x plus h, that's in the blank. And then for the second one, you just get an x. So you get something like this. Rewriting it a little bit simpler, you're going to get one over x plus h, x plus h squared minus one over x squared all over h. So in this situation, what we have is we have a bunch of nested fractions. Sometimes we call it compounded fractions or complex fractions. We have fractions inside of fractions. We have to deal with that. So you're going to notice that with these fractions, there's one big fraction bar. So that's like the mother bar. And then you're going to have some little fractions. These are little babies. We want the babies to move out. So what we're going to do is we're going to identify the least common denominator of the babies. In which case we then see the least common denominator is going to be x plus h squared times x squared. We're going to times the top and bottom by the LCD. We're going to time the top and bottom of the mother fraction by the LCD of the babies, okay? Now on the top, since you have a difference, you're going to distribute. And so this makes it look like something like the following. You're going to end up with an x squared times x plus h squared over x plus h squared. You're going to subtract from that x squared over x plus h squared over x squared. And then this sits above h times x squared times x plus h squared. Now in the denominator, do not factor that thing out because we don't want denominators multiplied out. We want to leave a factor. Don't multiply out the denominator. Leave it factored. Denominators should always be factored. But you're going to see if you look at the baby fractions in the numerator, there's some cancellation that happens. X plus h squared cancels, x squared cancels. So now all the baby fractions have moved out of mommy's fractions basement. Then we have x squared minus x plus h squared sitting above h times x squared and x plus h squared. While we don't multiply out denominators, we do want to multiply out the numerator. So you're going to need to foil up that x plus h quantity squared there. And upon doing so, you're going to get x squared plus two x h plus h squared sitting above that our denominator, h x squared x plus h squared. For which you can see that the x squares cancel. You get these x squares that cancel out. Then everything in the numerator is now negative. I'm just going to put it in front of the fraction. So you have a negative like this. What's left behind? You have a two x h plus h squared over h times x squared times x plus h. You'll notice that the numerator, now everything's divisible at h, right? You have an h right here and you have an h right here. Let's factor it out. Our goal is to factor out an h from the numerator so we can cancel with the h in the denominator. That's our goal. H here stands for hate. We hate the h in the denominator so we have to get rid of it. I know that's a violent image there. I apologize, but you'll remember it. We hate the h. So factor out the h so you get two x plus h left in the numerator and then you have this h times x squared times x plus h squared. For which then the h on the top cancels with the h in the bottom. And we see now that the average rate of change of our function has the formula delta y over delta x is equal to negative two x plus h over x squared times x plus h. So this gives us a formula for the average rate of change for which we could then insert any interval a comma a plus h. We can find the slope of that secant line, the average rate of change. But we're also interested in the instantaneous rate of change, the so-called derivative dy over dx for which we get the derivative here by setting h equal to zero. We see what happens as h goes to zero. For which case you're gonna get negative two x plus zero over x squared times x plus zero squared. Simplifying that we get two x over x to the fourth in the denominator for which that negative two x over x to the fourth. So we get negative two over x cubed as our instantaneous rate of change for which we can then plug in any value in for x and get the instantaneous rate of change at that moment. These fractional ones can be a little bit tricky. So I wanna do yet another example. Let's find the average rate of change and the instantaneous rate of change of the function x over x plus one. So when we start off, we always have to start off with the average rate of change, right? Cause we can't plug in h equals zero yet. We have to do that at the very end. So we have to take f of x plus h. So that's gonna give us x plus h over x plus h plus one. And then we subtract from that x over just x plus one. Like so, so each of the x's in the original formula I replaced with an x plus h. And then we have to subtract f of x and this all sits above h. The approach is gonna be the same. We have to identify the LCD of the baby fractions, which is nearly always good in these situations. It's nearly always just gonna be the product of the two. So we get x plus h plus one times x plus one, right? So we need to times the top and bottom by that LCD, all right? So then we get x plus h times x plus h plus one times x plus one, this sits above x plus h plus one. Hesitate to multiply out the numerator, right? We always wanna wait to multiply until it's actually beneficial to us. Then for the second fraction, we get x times x plus h plus one times x plus one all above x plus one. And this then sits above my denominator for which the denominator always leave it factored. h times x plus h plus one times x plus one. That's gonna be, that's where we are right now. So cancel will be canned. x plus h plus one cancels here, x plus one cancels right there. And so then with the simplified fractions, what do we have? We have an x plus h, we have an x plus one. And then we subtract from that x times x plus h plus one. And this sits above our denominator, leave it factored. h times x plus one times x plus h plus one. Now in the numerator, we do have to expand things in combined like terms. So FOIL and multiply is what's gonna be necessary. You have an x plus h times an x plus one. If you FOIL it out, you'll get x squared plus x plus xh plus h. And then you just subtract from that, just distribute the x there. You're gonna get x squared plus xh plus x. And this sits above our denominator, which is along for the right right now. All right. In which case now we need to combine terms that we can, the x squares cancel, the x cancels, and the x plus h cancels. Wow, that really decimated that group. The only thing who survived was the h there. So we have h over h times x plus one times x plus h plus one. Like so, for which then we can see the h on top cancels with the divisor of h on the bottom. So we now see that the simplified average rate of change is equal to one over x plus one times x plus one plus h or something like that. That's the simplified average rate of change. If we want the instantaneous rate of change, right, we set h equal to zero. We end up with one over x plus one times x plus one plus zero. That is this becomes in the simplified form one over x plus one quantity squared. Like so.