 In this example, let's compute or I should say simplify another difference quotient involving this time the function g of x equals the square root of x squared plus nine. How do you simplify a difference quotient that involves a square root of some kind? Well, let's look at our difference quotient right here. We have f of t minus f of zero over t minus zero. So t is some variable that's as the name suggests is able to vary. And then zero is some fixed number. How do we compute this difference quotient right here? Well, we have to compute f of t. What's f of t? And I can see that there might be confusion like what the heck is f there? You might be like, oh, look at that butterfly in the corner. And when you're no one's looking, you'll notice, oh, f, of course, was f of x equals the square root of x squared plus nine, wasn't it? G of x, who's that? I have no idea. So we want to compute the difference quotient of f of t minus f of zero over t minus zero. f of t just means replace all of the x values with t's. And so we're going to get the square root of t squared plus nine. Then subtract from that f of zero, f of zero would be the square root of nine, plug it in x equals zero there. And then the denominator, you're going to get t minus zero, minusing zero from anything. We'll just give you back the other number. You get back a t. So we get the square root of t squared plus nine minus the square root of nine is a three. So we get the following. And so we get this, which you might argue this is simplified, but like we've seen in previous difference quotients, in the previous setting and the current setting as well, if the denominator is a t, that means t can't equal to zero. But when it comes to rates of change, I actually want t to equal zero. How do we get past that obstacle? And it turns out we have to simplify this thing algebraically. Now we're going to bring up all of these wonderful skills we learned in a previous math class like say math 1010. In this case, in order to simplify this difference quotient farther, we need to rationalize. Now in math 1010, they typically, you typically learn to rationalize the denominator. We're going to do something extremely, extremely counterculture right now. We are going to rationalize the numerator. This is unheard of. The hippies are running wild right now. We are going to rationalize the numerator. Now remember how you do that is when you look at your, you have the square root minus or plus some number, right? What you're going to do is you're just going to look at that number, sorry, the sign that separates the square root. And you're just going to switch it from a negative to a positive or positive to negative, right? You're just going to be that person who just always says the opposite, right? You just have to be argumentative, switch the negative to a positive. And so this is called the conjugate of that square root expression right there, the conjugate. You're going to multiply the top and bottom by the conjugate. You get T squared plus nine plus three right there. And so in the numerator, I want you to foil this thing out, you're going to foil out the, you get a square root times that times three, three times that times that. Do all of the possibilities. You're going to get a square root of T squared plus nine quantity squared. You're going to get a plus three times the square root of T squared plus nine. You're going to get a minus 3 times the square root of t squared plus 9 and then finally you're going to get a minus 9 That's what you get in the numerator now in the denominator. What do you get? You're going to get t Times the square root of t squared plus 9 plus 3 that is you're not going to multiply out the denominator no multiply Denominator we've talked about this before it's actually better for a fraction to keep the denominator factor You're not doing yourself any favor by multiplying it out. So leave it factored I'm giving you permission not to do that step So don't do it because it's not going to be it's not going to be effective use of your time Well, why did we multiply at the numerator then well? There actually is going to be some benefit of doing that notice some simplifications that happen For example when you square the square root these are inverse operations. They cancel out So you end up with t squared plus 9 Some other things you have a 3 square root of t squared plus 9 You have a negative 3 square of t squared plus 9 those things are opposite, but equal They're going to cancel out. They're gone That's the reason why we multiply by the conjugate when you multiply by the conjugate and you foil out the outside and inside terms They're going to cancel and then the only other thing there is a minus 9 and then the denominator which we have refused We refused we took a blood oath our brothers Brotherhood we belong to our fraternity our sorority has forbidden us to multiply out the denominator And we will not let down our brothers or sisters here So the numerator though we have a t squared plus 9 minus 9 the plus minus 9 cancel I can do that and you're left now with a t squared over T times the square root of t squared plus 9 plus 3 You'll notice down the numerator is only thing left is a multiple of t So does the denominator has a multiple of t those multiples of t will cancel and the simplified difference quotient will be t over the square root of t plus 9 plus 3 and So now let's consider is this difference quotient better than what we start with You'll notice that the denominator no longer has a multiple of t in it if you were to plug in t equals 0 What's going to happen? You get a 0 on top you're going to end up with the square root of 9 plus 3 which ends up being 0 over 6 Which is just 0 notice that's not undefined right when you started this game You had a 0 over 0 0 0 is not a number it does not exist But then because of simplification we end up with a 0 over 6 which is a number it's 0 So because we were able to algebraically simplify it We actually could put a hole or that is we could patch the hole that was in our average rate of change Function and that's what gives us the instantaneous rate of change We simplify difference quotients so that we can set the denominator equal to zero It's a beautiful process very difficult very challenging, but we do hard things We learn algebra to do hard things we go to college to do hard things We don't run away from a challenge. We embrace it. We accept it and we conquer it That is what college is all about overcoming difficult things such as simplifying difference quotients It's a great thing and we'll do another example in the next video. Take a look for that link right now