Quotient rule | Derivative rules | AP Calculus AB | Khan Academy

What we're going to do in this video is introduce ourselves to the Quotient Rule, and we're not going to prove it in this video. In a future video, we can prove it using the Product Rule, and we'll see it has some similarities to the Product Rule. But here, we'll learn about what it is and how and where to actually apply it.

So, for example, if I have some function f of x and it can be expressed as the quotient of two expressions, so let's say U of x over V of x, then the Quotient Rule tells us that f prime of x is going to be equal to... and this is going to look a little bit complicated, but once we apply it, you'll hopefully get a little bit more comfortable with it.

It's going to be equal to the derivative of the numerator function, U prime of x, times the denominator function V of x, minus the numerator function U of x, times the derivative of the denominator function V prime of x. This already looks very similar to the Product Rule. If this was U of x times V of x, then this is what we would get when we took the derivative if this was a plus sign.

But this is a minus sign. We're not done yet; we would then divide by the denominator function squared V of x squared. So let's actually apply this idea.

Let's say that we have f of x is equal to x squared over cosine of x. Well, what could be our U of x and what could be our V of x? Well, our U of x could be x squared, so that is U of x, and U prime of x would be equal to 2x. Then this could be our V of x, so this is V of x, and V prime of x, the derivative of cosine of x with respect to x, is equal to negative sine of x.

Then we just apply this. Based on that, f prime of x is going to be equal to the derivative of the numerator function—that's 2x—right over here. So it's going to be 2x times the denominator function, V of x, which is just cosine of x, minus the numerator function, which is just x squared, times the derivative of the denominator function. The derivative of cosine of x is negative sine of x.

So, all of that over the denominator function squared, so that's cosine of x squared. I could write it, of course, like this. Actually, let me write it like that just to make it a little bit clearer.

At this point, we just have to simplify. This is going to be equal to... let's see, we're going to get 2x times cosine of x, minus a negative, which is a positive: plus x squared times sine of x, all of that over cosine of x squared, which I could write like this as well. And we're done!

You could try to simplify it; in fact, there are not obvious ways to simplify this any further. Now, what you'll see in the future—you might already know something called the Chain Rule, or you might learn it in the future—but you could also do the Quotient Rule using the Product and the Chain Rules, which you might learn in the future. But if you don't know the Chain Rule yet, this is fairly useful.