 In this video, we provide the solution to question number 14 for practice exam 1 from math 1050. We're given the function f of x equals x cubed, and we're asked to set up and simplify the difference quotient. All right, setting up is a pretty big deal, so let's start with that. So what is f of x plus h minus f of x all over h, all right? Well f of x plus h means that in the formula of f, everywhere you see an x, you replace it with an x plus h uniformly. So that for us, x plus h would mean x plus h cubed. And then we subtract from that f of x. f of x, the formula was given to us, it's just x cubed, in which case we get this right there, and then this will sit above an h. This is how we set it up. That's important. We get a lot of points just from setting this thing up correctly. We'll have to simplify, and I'll do that in just a second. I want to make a very common, point out a very common mistake here. It's very commonly mistaken that f of x plus h equals f of x plus h. And if you're listening to the audio here, it makes sense why that might be a mistake here. It sounds the same thing when you say it, but it's the order of operations that matter. The plus h is inside the function, it's not outside the function. So in fact, that's not the case. f of x plus h doesn't mean you take the function f of x and add h to it. Now if this were true, if f of x plus h was what it meant to have an x plus h right here, notice your difference quotient would always do the following. f of x plus h minus f of x over h. Well, if you could then replace this f of x plus h with this expression right here, the f of x plus h, right? Then notice that irrelevant to the formula of f of x, the f of x is canceled. So you end up with h over h, and then the h is canceled, you end up with a 1. And so if you get a difference quotient and it simplifies to a 1, I'm going to give you a 99% chance that it is wrong, and then you probably made this mistake. You don't want to make this mistake because this mistake forfeits nearly all of the points on this problem. So be cautious of that. f of x plus h means you put x plus h inside of the function formula instead of the x that was already there. Now when we look at x plus h cubed, we have to multiply that out. If you know the binomial theorem, that speeds things up a lot, but we will go without that knowledge here. We're going to be going as if this is x plus h times x plus h times x plus h. So as we foil this thing out, what is x plus h squared? You'll get an x times an x, which is an x squared. You'll get an x times an h, you'll get another h times x, so you get 2xh, like so, and then you'll get an h times h, which is h squared. Like so, there's still an x plus h right here, because we haven't done the second one yet. Let me squeeze in my negative x squared and negative x cubed, excuse me, all over h. Our goal is to grid the h at the bottom, but we have to expand the numerator before we can do that. So the next thing to do is to multiply out the x squared plus 2x plus h squared with the x plus h, like so. And so we just do this step by step, right? We get x squared times the x times the h. This is going to give us x squared times x is x cubed. x squared times h is going to be x squared h. The next goal is then to distribute the 2xh on the two of these pieces, for which we take 2xh times x. We're going to end up with a 2x squared h. You'll notice that these are actually like terms, we'll combine them in a second. 2xh times h is going to be a 2xh squared. Now the last one to do is distribute the h squared onto both of these terms. So next we're going to get h squared times x, which gives us an xh squared. You'll notice that these are also like terms, which we'll combine in just a moment. And then lastly, you get h squared times h, which is an h cubed, like so. And then don't forget the other things that were along for the journey. We have a negative x cubed over h, like so. So inside the parentheses, there are some terms that are alike. Let's add them together. We ended up with an x cubed. We have a 3x squared h. We have a 3xh squared. And then there's an h cubed, like so. This is exactly what the binomial theorem would have produced for us a lot faster. But if we don't know the binomial theorem, not a big deal. We're able to do it just by usual foil. And so now notice what happens here is that in the numerator, we have an x squared. We have a minus x squared. Those are going to get each other, so they cancel out like so. Then, now that we've reduced the numerator, we have this 3x squared h plus 3xh squared plus h cubed all over h. I'm going to bring this down a little bit. Notice everything in the numerator is divisible by h. We have an h here, we have an h here, we have an h here. We can factor out that h. So you fact out the h what's left behind, 3x squared plus 3xh plus h squared, like so. This all sits above an h. And it's this factor of h in the numerator that cancels with the factor of h in the denominator. So they cancel out. We could not have canceled out the h any earlier than that. And so the simplified version of the difference quotient is 3x squared plus 3xh plus h squared. This one was definitely on the harder side of how simplifying a difference quotient can go.