Quotient Rule f(x)=(x2-4)/(x2+4) Hi everyone, welcome back. We’re going to be working on some quotient rule problems. Quotient rule is like product rule, chain rule, or reciprocal rule. It’s just a two-rule that helps you take the derivative of different kinds of functions. So, I’m going to go ahead and write it out so that we have for reference here, and I will post it to the formulas page as well. And all it says is that when you have a function like this, you’re trying to take the derivative of a function with something in the numerator and something in the denominator. The way that you do it, it’s going to end up looking like this, f1(x), the derivative of the top times the bottom minus the top times the derivative of the bottom divided by the bottom, g(x)2. So, it gives you an easier from to follow. So the problem that we’re going to be applying this two is actually f(x) = (x2 - 4)/(x2 + 4). So, we will use the quotient rule, apply it to this function and be able to take the derivative. So I’m going to go ahead say (x), the derivative, and we’re just going to follow this step-by-step. So the first thing is the derivative of the top because here’s the top of effect so we need to take the derivative of the top. So the derivative of x2 - 4 is just going to be 2x, and then we need the bottom. So g(x), x2 + 4, and then minus. So minus the top, x2 - 4 and then the derivative of the bottom, g(x). So the derivative of the bottom, that is also 2x. So we have that, and then we do the bottom, g(x)2. So we have x2+4 which is the bottom but then we’ve got a square root because that’s part of the formula. So, you could leave this as your answer but it would probably be best considering that there are so many similar terms in this function to simplify it. So we’ll go ahead and do that for you guys. So the derivative is going to be 2x × x2 = 2x3+8x-2x3+, there’s a plus because we’ve got a minus and minus, we multiply them together and we’ve got a +8x/x2+42. So now we have 2x3-2x3, those go away and we’ve got 8x+8x. So it’s actually going to be—I’m going to go ahead and write it over here. The answer is actually going to be 16x/x2 + 42, and that’s the answer.