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# Quotient rule

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.B (LO)
,
FUN‑3.B.2 (EK)

## Video transcript

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 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 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 when we took the derivative if this was a plus sign but this is here a minus sign but 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 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 ru of X could be our x squared so that is U of X and u prime of X would be equal to 2x 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 new Reiter function that's 2x right over here that's that there so it's going to be 2x 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 going to get two x times cosine of X 2 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 it's like there's no not 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 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
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