 So a key idea in calculus is the idea of a difference quotient, and this emerges as follows. Suppose I have some function f of x. I might want to consider the value of that function in two places, at x and at some other place, x plus h. And we have a couple of differences we can talk about. First off, we can talk about the difference between the function values. And this difference is just the change in our function from x to x plus h. So going back to our notions on rates of change, this is just the change in the function value. And then we also have the difference between the x values, where we end up, x plus h, and where we start at x. And once we have these two changes, we can talk about the ratio of change, the rate of change, which is the quotient between the two, and it's going to be what our difference quotient is. So that's the change in our function value over the change in our x values. And that simplifies a little bit, since x plus h minus x is just equal to h. We have this simplified form of what our difference quotient is. So let's take a look at an example. Suppose I have f of x equals x squared plus 3x minus 5. We can find our difference quotient. And we need to, first of all, find f of x and f of x plus h. And this just involves substituting the arguments x and x plus h into our function definition, although sometimes that can cause problems. So here's one suggested way that you might approach this. So we have our definition of the function. And what we might do is to drop out all of our variables, all of our independent variables, and replace them with an empty set of parentheses. So I'm just going to drop the x out. So x is gone, x is gone, and replaced with an empty set of parentheses, still squared. x is gone, replaced with an empty set of parentheses, still times 3. And the minus 5, nothing happens to. And the key idea here, as with any function expression, is whatever you put in one set of parentheses has to be the same in all of the other sets of parentheses. So what do I want? Well, I already know what f of x is. What I want to know is what f of x plus h is. So I'm going to drop an x plus h in here and in the other two places. So I'm going to drop that in like that. And I'll do a little bit of algebra. I'll expand out those expressions. And now I can just write down the difference quote. And if I want to make my life easier, I'm going to do some simplification. So what do I need? I need f of x plus h. There it is. I need f of x. There it is. And then I need to subtract them and divide by h. So I'll write that down. f of x plus h minus f of x over h. This is our f of x plus h. That's f of x. And we do a little bit of expansion. So we'll subtract, distribute that minus among everything. And we can simplify. We have a couple of things that will cancel out. x squared and x squared, 3x and minus 3x, minus 5 and plus 5. Those all go away. And what we have left is this expression. And here's a useful check when you're working with a polynomial quotient. Notice that each of the three terms here has a factor of h. And what that means is I can remove that factor of h from those terms. And I can divide the h out. I can cancel it out with the qualification that I can't actually let h be equal to 0. And so at the end of it, my difference quotient for this function, 2x plus h plus 3.