 This is going to be our final video for lecture 6. And I want to do one more simplification of a difference quotient. This time, take the rational function f of x equals 1 over x squared and consider the difference quotient f of x plus h minus f of x over h. We know that there's a problem when h goes to 0. By simplifying the difference quotient, we are then able to remove the problem in the domain at h equals 0. We can extend the domain to be h equals 0 by simplifying this difference quotient. So what is going to be x of x plus h? Well, this just means plug in x plus h in for x here. You get 1 over x plus h squared. Then what about f of x? Well, let's just 1 over x squared. We can see it right here. Here's f of x. I guess you can't see it if I draw over it all the time, but you get what I'm trying to say there. And then we have an h right there. All right. So at this moment, if you would try to plug in h equals 0, what you're going to see is a 1 over x squared minus 1 over x squared over 0. That already should give us pause because we have 0 on the denominator. But you'll notice it actually simplifies to be 0 over 0. When you get something like 0 over 0, that actually means calculus. This is like shining the bat signal into the sky because you want Batman to come to save the day. If you shine the symbol 00 onto the clouds in the moonlight, this means the mathematician will come to try to save the day. We want to save the day because 0 over 0 is calling us. This gives us hope that we can simplify this difference quotient, but how do we do it? If you try to multiply out x plus h squared, I will stop you right there because we don't multiply out denominators. Instead, we see these baby fractions inside of a big fraction. We don't like that. We want to simplify this compounded fraction. And so in order to do that, look at the denominators of all the baby fractions. You have an x plus h squared and an x plus and then just an x squared. This tells us that the least common denominator amongst the babies will be the product of the two x squared times x plus h squared. You are going to multiply the numerator of the big fraction by the LCDs of the little fractions and you're going to do that to the denominator as well. Now in the numerator, you're going to distribute these things through. We'll see in a moment exactly why we do that. And the denominator, leave things alone because we do not multiply out denominators. Not going to do it, you cannot force me. With the first one, you'll end up with x squared times x plus h squared over x plus h squared. For the second baby fraction, you get x squared over x plus h squared over x squared over h times x squared times x plus h squared, like so. Now you'll notice in the numerator, if you look at the baby fractions, oh, there's an x plus h squared, x plus h squared, they cancel out. With the second baby fraction, if there's an x squared and x squared, they cancel out. And so now the baby fractions all moved out of mom's basement and we're left with x squared minus x plus h quantity squared. This sits above our very unfactored denominator. It's a utopian society when you don't factor your denominators there. H times x squared over x plus h squared. Now, now at this moment, we're going to foil the x plus h squared in the numerator because you'll notice it's no longer in a denominator. So there's no reason not to expand things. If you multiply out the x plus h squared, you'll end up with an x squared plus two x h plus h squared over h times x squared times x plus h squared. You'll notice as you distribute that negative sign, there's now an x squared and a negative x squared. They cancel out. Who didn't get canceled? You're gonna have a negative two x h minus h squared. You'll notice they're both negative now and that's because we distributed this negative sign. And this sits above h times x squared times x plus h squared. Notice now everything, it's like magic, right? Math and magic. Everything in the numerator is divisible h. Factor it out. It's like four ordained for all of this to happen here. You're gonna get negative two x minus h times h all over h times x squared times x plus h squared. We're now in a situation where we can cancel out the h in the denominator. And now we have our simplified difference quotient, negative two x minus h over x squared times x plus h squared. And now in this simplified difference quotient, you will notice there is no obstruction to plug in an x equals zero. Go ahead, do it. I dare you. If you plug in h equals zero, you will not get zero in the denominator. You'll actually end up with x to the fourth, which there's no problem with that. And so we are able to fix the problem about h being zero by simplifying the difference quotient. So in this section, in this lecture six, we practice a lot on simplifying these difference quotient. This is a very important calculation. One of the hardest things we're actually gonna do this in this series, right? And you're like, lecture six, we're doing some of the hardest stuff. Well, yeah, we're gonna start off with some of the hard stuff. And that's actually gonna make the rest of the series a lot, lot easier, I think. Be patient with yourself. Don't make the same mistake that Daniel Son makes, right? Be patient. Sand the deck, paint the fence. If you're patient with me, we'll see that the investment of time right now will pay dividends later on. And that brings us to the end of lecture six. Stay tuned for lecture seven, which we'll do next time. We're gonna talk about inverse functions, which is a very important calculation to do. And hopefully we'll see you then. Bye everyone.