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# Differentiating rational functions

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

## Video transcript

let's say that Y is equal to five minus three X over x squared plus three X and we want to figure out what is the derivative of Y with respect to X now it might immediately jump out at you that look-look Y is being defined as a rational expression here as the quotient of two different expressions we could even view this as two different functions you could view this one up here as U of X so you could say this is the same thing this is the same thing as U of X over you could view the one in the denominator as V of X so that one right there is V of X and so if you're taking the derivative of something that can be expressed in this way as the quotient of two different functions well then you could use the quotient rule and I'll give you my little aside like I always do the quotient rule if you ever forget it it can be derived from the product rule and we have videos there because the product rule is a little bit easier to remember but what I can do is just say look dy DX with if Y is just U of x over V of X I'm just going to restate the quotient rule this is going to be this is going to be the derivative of the function in the numerator so d DX of U of x times the function in the denominator times V of X - - I'll do the - the function in the numerator U of x times the derivative of the function in the denominator times D DX V of X and we're almost there and then over over the function in the denominator squared the function in the denominator squared so this might look messy but all we have to do now is think about well what is the derivative of U of X what is the derivative of V of X and we should just be able to substitute those things back into this this expression we just wrote down so let's do that so the derivative with respect to X of U of X of U of X is equal to let's see 5 minus 3x so the derivative of 5 is 0 the derivative of negative 3x well that's just going to be negative 3 that's just negative 3 if any of that looks completely unfamiliar to you I encourage you to review the derivative properties and maybe the power rule and now let's think about what is the derivative with respect to X derivative with respect to X of V of X of V of X well derivative of x squared we just bring that exponent out front it's going to be 2 times X to the 2 minus 1 or 2 X to the first power or just 2 X and then the derivative of 3x is just 3 so 2x plus 3 and now we know everything we need to substitute back in here the derivative of U with respect to X this right over here is just negative 3 V of X this we know is x squared plus 3x we know that this right over here is V of X and then U of X we know is 5 minus 3x 5 minus 3x the derivative of V with respect to X we know is 2x plus 3 2x plus 3 and then finally V of X we know is x squared plus 3x so this is x squared plus 3x and so what do we get well we are going to get and it's going to look a little bit hairy it's going to be equal to negative I'll focus so first we have first we have this business up here so it's negative 3 times x squared plus 3x so I'm just going to distribute the negative 3 so it's negative 3x squared minus 9x and then from that we are going to subtract the product of these two expressions and so let's see what that what's what is that going to be well we have a 5 times 2x which is 10x of 5 times 3 which is 15 we have a negative 3x times 2x so that is going to be negative as 6x squared minus 6x squared and then a negative 3x times 3 so negative 9x and let's see we can simplify that a little bit 10x minus 9x well that's just going to leave us with X so if we 10x minus 9x is just going to be X and then in our denominator we're almost there in our denominator we could just write that as X plus 3x x squared plus 3x squared or if we want we could expand it out I'll just leave it like that x squared plus 3x squared and so if we want to let's just simplify or attempt to simplify this a little bit it's going to be negative 3x squared minus 9x and then let's see you're going to have a negative minus X minus X and then minus 15 and then minus negative 6x squared so plus 6x squared all of that all of that over x squared plus 3x squared or x squared plus 3x squared I should say it that way now let's see this numerator I can simplify a little bit negative 3x squared plus 6x squared that's going to be positive 3x squared and then we have we have other than orange we have negative 9x minus an X well that's going to be minus 10x minus 10x and then we have minus 15 so minus 15 and so there you have it we finally have finished this is all going to be equal to this is all going to be equal to 3x squared minus 10x minus 15 over x squared plus 3 x-squared and we are done