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## AP®︎/College Calculus AB

# Quotient rule

AP.CALC:

FUN‑3 (EU)

, FUN‑3.B (LO)

, FUN‑3.B.2 (EK)

Introduction to the quotient rule, which tells us how to take the derivative of a quotient of functions.

## Video transcript

- [Instructor] 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. Its 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. Do that in that blue color. U of X. Times the derivative of
the denominator function times V prime of X. And 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 if we took the derivative this was a plus sign. But this is here, a minus sign. But were not done yet. We would then divide by the denominator function squared. V of X squared. So let's actually apply this idea. So 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 our X squared. So that is U of X and U prime of X would be equal to two X. And 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. And then we just apply this. So based on that F prime of X is going to be equal to the derivative of the numerator function that's two X, right over
here, that's that there. So it's gonna be two X times the denominator function. V of X is just cosine of X times cosine of X. Minus the numerator function which is just X squared. X squared. Times the derivative of
the denominator function. The derivative of cosine of X is negative sine X. So, negative sine of X. All of that over all of that over the denominator function squared. So that's cosine of X and I'm going to square it. I could write it, of course, like this. Actually, let me write it like that just to make it a little bit clearer. And at this point, we
just have to simplify. This is going to be equal to let's see, we're gonna get two X times cosine of X. Two X cosine of X. Negative times a negative is a positive. Plus, X squared X squared times sine of X. 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's not an obvious way
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 rule that you might learn in the future. But if you don't know the chain rule yet, this is fairly useful.