Worked example: Quotient rule with table | Derivative rules | AP Calculus AB | Khan Academy

Let F be a function such that f of 1 is equal to 3. Frime of 1 is equal to 5. Let G be the function G of x is equal to 2x cubed. Let capital F be a function defined as so capital F is defined as lowercase f of x divided by lowercase G of x. And they want us to evaluate the derivative of capital F at x equal to -1.

So the way that we can do that is let's just take the derivative of capital F and then evaluate it at x equal to 1. The way they've set up capital F, this function definition, we can see that it is the quotient of two functions. So if we want to take its derivative, you might say, well, maybe the quotient rule is important here.

I'll always give you my aside: the quotient rule. I'm going to state it right now, and it could be useful to know it, but in case you ever forget it, you can derive it pretty quickly from the product rule. If you know it, the chain rule combined, you can get the quotient rule pretty quick. But let me just state the quotient rule right now.

So if you have some function defined as some function in a numerator divided by some function in the denominator, we can say its derivative— and this is really just a restatement of the quotient rule— its derivative is going to be the derivative of the function in the numerator, so D/Dx F of x times the function in the denominator, so times G of x, minus the function in the numerator, minus F of x not taking its derivative times the derivative of the function in the denominator, D/Dx G of x, all of that over— so all of this is going to be over the function in the denominator squared— so this G of x squared.

And you could use multiple different types of notation here; you could say instead of writing this with the derivative operator, you could say this is the same thing as G prime of x. Likewise, you could say, well, that is the same thing as F prime of x.

Now we just want to evaluate this thing, and you might say, well, how do I evaluate this thing? Well, let's just try it. Let's just say, well, we want to evaluate F prime when x is equal to 1.

So we can write F prime of -1 is equal to— well, everywhere we see an x, let's put a negative 1 here. It's going to be F prime of 1 times G of 1 minus F of 1 times G prime of 1, all of that over G of -1 squared.

Now can we figure out what F prime of 1, F of 1, G of -1, and G prime of -1 are? Some of them they tell us outright; they tell us F and F prime at negative 1. For G, we can actually solve for those.

So let’s see, if this is— well, let’s just first find G of 1. G of 1 is going to be 2 * (-1) to the 3rd power. Well, -1 to the 3rd power is just -1, so this is -2.

And G prime of x— I’ll do it here. G prime of x, which use the power rule, bring that 3 out front; 3 * 2 is 6. x decrement that exponent 3 - 1 is 2, and so G prime of -1 is equal to 6 * (-1) squared. Well, (-1) squared is just 1, so this is going to be equal to 6.

So we actually know what all of these values are now. So first we want to figure out F prime of 1— well, they tell us that right over here: F prime of 1 is equal to 5.

So that is 5. G of -1— well, we figured that right here: G of -1 is -2. So this is -2. F of 1— let me, so F of 1— they tell us that right over there: that is equal to 3.

And then G prime of -1— let me just circle it in this green color— G prime of -1 we figured it out; it is equal to 6, so this is equal to 6. And then finally, G of -1 right over here, we already figured that out; that was equal to -2.

So this is all going to simplify to— so you have 5 * (-2), which is -10, minus 3 * 6, which is 18. All of that over -2 squared— well, -2 squared is just going to be positive 4, so this is going to be equal to -28 over positive 4, which is equal to -7.

And there you have it. It looks intimidating at first, but if you just say, okay, look, this is— I can use the quotient rule right over here. Then once I apply the quotient rule, I can actually just directly figure out what G of -1, G prime of -1, and they gave us F of 1 and F prime of -1. So hopefully, you find that helpful.