 So, where does all of this talk about limits lead us? It leads us to the idea of a derivative, so we're ready to introduce the definition of a derivative. It's actually relatively simple, suppose I have any function f of x, then there's this thing that I'm going to call the derivative of f of x, and I'm going to write that f prime of x, that little hash mark is red prime, and I'm going to define it in the following way. f prime of x, the derivative of f of x, is going to be defined as the limit as h goes to zero of f of x plus h minus f of x over h. Now, remember limits don't always exist, so we do have to add in, in our fine print, that this derivative only exists as long as the limit actually does. And something that's worth noting, this expression here should be very familiar. It's just the difference quotient of f of x, so another way of looking at the derivative, it's just the limit of the difference quotients. Now, again, because the limit doesn't always exist, we do want to use an adjective to describe a function for which the derivative does exist, and so we say that if the limit exists, then our function is differentiable. And in one of the classic displays of how good mathematicians are coming up with new names for things, if we're in a situation where this limit does not exist, in other words, the limit as h goes to zero of a difference quotient, does not actually have a value, then we say that f is not differentiable. Now, while mathematicians are really bad at coming up with good names for things, they're really good at coming up with notation. And one of the problems, and one of the good things, because it cuts both ways, that we write into is there's actually two slightly different ways of expressing the derivative. This method, f prime of x, which is called prime notation, is actually fairly common because it is a nice compact way of expressing the idea of the derivative. However, there's another form of notation that's called differential notation, and differential notation looks something like this. What we have is our f prime of x, the derivative of our function, and in place of those primes, our differential notation, uses something that looks an awful lot like a fraction. It says d over dx, and there's our function f, which is being differentiated. And again, while this expression, d over dx, looks like a fraction, it's important to remember it's not actually a fraction, although one of the reasons that differential notation persists is that it's actually pretty useful. There will be times when it is convenient to think about this differential notation as corresponding to a fraction. And we'll talk about that. So let's take a look at a couple of derivatives. So let's start off with a nice simple polynomial function, f of x equals x squared plus 3x minus 5, and I want to use the definition defined f prime of x. Now a little bit later on, we'll see how to find derivatives, the easy way, there'll be nice simple formulas for it. And if we are just looking for f prime of x, those nice simple formulas will work. However, this question specifically asks us to use the definition to find f prime of x. And because it asks us to use the definition, there is no other way of solving this problem than to use the definition. So we'll go ahead and write down our definition. f prime of x is the limit as h goes to 0 of our difference quotient. And well, we've actually already found the difference quotient for x squared plus 3x minus 5. And just to go over those steps fairly quickly here, I have my limit of the difference quotient. I'll substitute x plus h every place I see an x in my expression. I have my function. And at this point, I'll do a little bit of algebra. I'll expand things out. I'll multiply this out. I'll use the distributive property that gives me something like that. I'm subtracting a whole bunch of stuff, so I'll distribute the minus. And I have like terms I can collect. x squared minus x squared h squared, let's see, 3x minus 3x minus 5 plus 5. I'll collect like terms and get rid of what I can. And look, every one of these terms here has a factor of h, so I can get the h out. And now I have a factor of h in the numerator, a factor of h in the denominator, and I can drop those. And I don't actually have to worry about that qualifier. We don't have to qualify this with an h can't be equal to 0 because I'm taking the limit. And the limit is asking what happens as h gets close to 0. And because I'm taking the limit, I don't actually care what happens when h is 0. So I don't need that qualifier this time, but now I do need to do a little bit of calculus. One thing that's worth noting, the calculus that we've done is here, writing down the definition of the derivative, and what we're about to do, which is actually taking the limit. Everything in between all those intermediate steps, those are algebraic steps. They're important to do, but they're not really the calculus. But they are the part that we often run into difficulties with because we have to keep track of everything. Algebra is mostly like bookkeeping. And if you want to do bookkeeping like Enron, then you probably won't do too well in getting the derivatives done correctly. But let's take a look at that. As h gets close to 0, 2x, nothing happens too. 3, nothing happens too. And h itself gets close to 0. So this expression, as h gets close to 0, gets close to 2x plus 3. And so joining beginning to end, the derivative of f of x, according to the definition, 2x plus 3. Well, that was fun. Let's do another one. So how about a rational function? So here's our function f of x equals 1 over 2x plus 5. Again, we're going to use the definition to find the derivative. So I've got to put down the definition. And again, it's just the limit of the difference quotient. So I'll substitute those values in. My function is 1 over 2x plus 5. Again, we actually found this difference quotient earlier. And so after all the dust settles, my difference quotient simplifies to this expression. And now to think about this, as h gets close to 0, minus 2 gets close to minus 2. 2x plus 5 gets close to 2x plus 5. And this expression here, the only thing that happens, is that this middle term here goes to 0. So this expression here goes to 2x plus 5. And so the limit as h goes to 0 of my difference quotient minus 2 over 2x plus 5 times 2x plus 5. And if you want to show off your erudition, which is defined as knowing the meaning of words like erudition, one of the things you can do is to note that this 2x plus 5 times 2x plus 5 is just 2x plus 5 quantity squared. But you don't really need to do that. Either answer is perfectly good. Now, how about that third difference quotient that we found for square root 3x minus 1? Let's go ahead and find our derivative. So my derivative again, I'll put down the definition. I'll go ahead and write down what our difference quotient is. And after all the dust settles, our difference quotient looks something like this. And so let's see. Again, as h gets close to 0, 3 stays 3. 3x minus 1 stays 3x minus 1. 3x plus 3h minus 1 becomes 3x minus 1. And so I have 3 over square root 3x minus 1 plus another square root 3x minus 1 as my limit of the difference quotient and again as my derivative of f of x.