 Let's look at a few examples of how to use the quotient rule in order to find derivatives. The quotient rule, of course, involves the quotient of two different variable expressions. Typically we think of the numerator as the f function and the denominator as the g function. So in order to use the quotient rule, we start in the denominator and keep that and we multiply by the derivative of the numerator. It's always minus, keep the numerator and multiply by the derivative of the denominator. And all you're doing here in this case for that is to use the power rule. Then we have it over the denominator squared. Of course, we can multiply out and simplify the numerator. We have to distribute the five, distribute the two x, you can give that a shot on your own. And if you were to do that for this, you should get negative 5x squared plus 4x minus 5. And we can simply keep that quantity as squared in the denominator. So let's look at another one. So we think of the numerator as our f function, denominator as our g function. Keep the denominator, start there. You can think of it as, I like to think of it as you go clockwise starting in the denominator. Maybe that will help you remember the order for everything. So there's your denominator. Now we multiply by the derivative of the numerator, just three. Again, it's always minus. Then keep the numerator and multiply by the derivative of the denominator. And we have that over the denominator squared. And once again, you can multiply out the numerator if you wish and simplify it, combine your similar terms. But that's about all you could do to simplify this one. Now the last example we're going to look at, we are going to do this two different ways. So let's do it first with the quotient rule. And then we'll talk about another way to do it. Now with the quotient rule, numerator and denominator, so let's go ahead and start that. So remember we start with the denominator, keep that. Then we multiply by the derivative of the numerator, so that would be 3x squared plus 8x minus. Now we keep the numerator and multiply by the derivative of the denominator. But notice derivative of three is simply going to be zero. So that quantity after minus pretty much zero is out. And then we have that over the denominator squared, which is nine. You could easily simplify this because you have the three up in the numerator and nine in the denominator. So we could make this 3x squared plus 8x all over three. All right, now let's be a little creative and think of this another way though. What if we were to have rewritten the original as maybe think of that one third, the three in the denominator as a constant one third, a constant multiplier. So if you sometimes take a couple minutes just to think about alternative ways to do this, you can actually make it a lot simpler. And I'm going to go ahead and distribute that one third because look at what happens now. So if we go and find our derivative, it's really now just a power rule problem. So we have one third times three, so that's of course one x squared plus we have eight thirds. Now notice how that really gives you the same as what the quotient rule yielded. Because if we take that 3x squared, put that over the three in the denominator, that gives you the x squared and the 8x over the three is the eight thirds x. So you get the same answer either way, but if you take a minute to just think about alternatives as to how you might be able to rewrite the original function, sometimes you can make it a lot easier on yourself to find the derivative you need.