 One of the reasons for taking a course called pre-calculus is to prepare you to take a course called calculus. And one of the important ideas you'll need in calculus is based on something called the difference quotient. And the difference quotient has a simple definition. Let f of x be a function of x. The difference quotient is f of x plus h minus f of x. There's our difference over h. And there's our quotient. And remember, definitions are the whole of mathematics. All else is commentary. At this point, you know everything you need to know about finding a difference quotient, and the only task ahead is actually finding it. And, fair warning, that's mostly algebra. From this point forward, we're going to be doing lots and lots and lots and lots and lots and lots of rather tedious algebra, all motivated by this central concept of a difference quotient. So, let f of x be x squared minus 3x plus 5. We'll find the difference quotient. Again, definitions are the whole of mathematics. So, let's pull in our definition of the difference quotient. So, let f of x be a function of x. Got it. And the difference quotient, we need to find f of x plus h and f of x. Well, f of x is easy. That's just given to us. And, if I want to find f of x plus h, remember, we can start off by dropping the independent variable and leaving an empty set of parentheses. Then, whatever we put in one set of parentheses has to go in all of them. And what we'd like in this first set of parentheses is x plus h. So, we'll put an x plus h in every set of parentheses. And let's go ahead and expand that. And we want our numerator to be f of x plus h minus f of x. So, let's go ahead and figure out what that is. Equals means replaceable. So, let's replace f of x plus h and f of x. And don't forget to use parentheses around our subterhent. And let's clean up some of the algebra. And we end up with this expression. Again, equals means replaceable. So, this is f of x plus h minus f of x. An important idea to keep in mind is that the final expression of the difference quotient of an algebraic function should not include a denominator of h. We should be able to do something that will cause this rational expression to simplify so that we're not dividing by h. So, this means we can and should simplify our difference quotient. So, it's a quotient and my only hope of being able to cancel out that h is to have the numerator expressed as a product, which means I need to factor. So, now my numerator is a product h times stuff over h. And so those h's can cancel out. And I'm left with my final expression of the difference quotient. What if our function is g of x equals square root of 8 minus x? We find g of x and g of x plus h. g of x is easy. That's what we've been given. g of x plus h, well, we'll start out by rewriting our function, replacing every appearance of x with an empty set of parentheses. Then, whatever goes in one set of parentheses has to go in all of them. And so we have our difference quotient. Or do we? Remember that the final expression of the difference quotient of an algebraic function should not include a denominator of h. So, since our expression still has an h in the denominator, we need to get the h out. Sad to say, the only way we can do that is to resort to a bunch of algebraic tricks. And that means we have to spend a little bit of time doing tricks and not actually doing math. And one of the tricks we can turn is to remember that if I multiply the sum and difference of square roots, I eliminate the square roots. So, since I have a difference of square roots in the numerator, what I'd like to do is I'd like to multiply it by the sum of the square roots. And we can, as long as we pay for it. In this particular case, because I don't want to change the expression that I have, I have the difference quotient. I need to multiply and divide by this sum of square roots. So I'll multiply numerator and denominator by that sum. And now we can do a little bit of algebra. Now, one of the hallmarks of being a good math student is how effectively you procrastinate. So here's an idea to keep in mind. The reason that we multiplied by this horrifying expression is so that we could get rid of these square roots in the numerator. On the other hand, the reason we're doing all of this is so that we can cancel out this h in the denominator. But we can only do that cancellation when numerator and denominator are products. Now, in the denominator, I have h times horrifying mess, and that's already a product. So I'd like to leave it in that form. Of course, the numerator is also a product, horrifying mess times horrifying mess. But it's a product that has a much simpler final form. So let's multiply out the numerator. And now we can simplify it. And now my numerator is just minus h. My denominator is a product that includes the factor of h, so we can simplify and get our final expression for the difference quotient. What if I have a rational function? Well, again, we can find m of x and m of x plus h. We can write down our expression for the difference quotient, but we need to get the h out. So we have this horrifying fraction. But remember, if you multiply a fraction by its denominator, the fraction becomes an integer. So if we multiply through by the product of our denominators, 2 times x plus h plus 3, and 2x plus 3, then we'll eliminate all of our fractions. So we'll buy a 2 times x plus h plus 3 times 2x plus 3, and we'll pay for it by dividing by the same quantity. And again, let's procrastinate efficiently. The denominator is a product, so let's leave it in that form. On the other hand, the whole reason behind multiplying the numerator by this frightening mess is to get rid of the fractions. So let's multiply by the fractions, then simplify, then expand and combine like terms. And now our numerator is also a product minus 2 times h, so we can cancel out the common factors and get our final expression of the difference quotient.