 The next type of derivative rule we'll look at is the quotient rule, which is what we use whenever our function corresponds to a quotient. And like the product rule, in general, the derivative of a quotient is absolutely not the quotient of the derivatives. Instead, it's a rather more complicated expression in general. The derivative of a function that is a quotient, we have our numerator function and our denominator function, the derivative is going to be the denominator, bottom, times derivative of the numerator, derivative of the top, minus numerator function, top, times the derivative of the denominator, derivative of the bottom, all over the square of the original denominator. And that's in our differential notation. Equivalently, if we want to use prime notation to express this rule, we have derivative of a quotient is equal to the denominator times derivative numerator, minus numerator derivative denominator, all over the square of the denominator. So as with the product rule, given some values of a function and some values of the derivative of the function, we can find the derivative of the quotient of the two functions. So here I have a bunch of values of functions and derivatives, and I want to find the derivative of the quotient. So let's see. It is a quotient, so we apply the quotient rule. First thing, let's set it down. A good way of memorizing this formula is to just write it down a few times. As you use it, you will remember it. So the derivative of the quotient, bottom derivative top, minus top, derivative bottom, all over bottom squared. So I want to find the derivative at three. So I'm going to evaluate all the functions of the derivatives at three. And now I can just fill them in from the given information. So g of three, we know what that is. F prime of three, we know what that is. F of three, we know what that is. G prime of three, we know what that is. And g of three again, we know what that is. And we can do a little bit of arithmetic, but anything beyond this, such as reducing to lowest common terms and all that other stuff, is probably unnecessary. Now how about something that looks like an actual function? So let's find the derivative of 3x plus 5 over square root of x. And again, a little bit of analysis at the beginning goes a long way at the end of the problem if we consider this function, 3x plus 5 over square root of x. If I try to evaluate this for a given value of x, I need to do the following steps. x times 3, add 5, hold it. Take x, find the square root, hold it. And then I'm going to divide those two values. So because the last thing I do in this function is divide, this is a quotient. And if I want to find the derivative, I'll apply the quotient rule. So let's see what does that mean? Well, I'll write down the quotient rule. Once again, a good way of remembering this rule is to write it down every time you have to use it. And after a while, this rule is memorized. So here I'm going to use the differential notation again for variety. I have bottom derivative top minus top derivative bottom all over bottom squared. And at this point, I just need to fill in the blanks. So here I have derivative of the quotient. So my numerator function, 3x plus 5. So I'll fill those in where they should be. My denominator function, square root of x. So I'll fill that in where it needs to be. And again, I haven't actually differentiated yet. I still need to find derivative 3x plus 5, derivative square root of x. So let's go ahead and do that. So derivative of 3x plus 5 is 3, derivative of square root of x. Remember, this is x to power 1 half. So the derivative is going to be 1 half x to power minus 1 half. And in principle, I could do a lot of simplification here. But in practice, I'm just going to do one thing. I'm going to evaluate the square root of x squared. And there's my simplification.