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# Manipulating functions before differentiation

AP.CALC:
FUN‑3 (EU)

## Video transcript

what I have listed here is several of the derivative rules that we've used in previous videos if these things look unfamiliar to you I encourage you maybe to not watch this video because in this video we're going to think about when do we apply these rules what strategies and can we algebraically convert expressions so that we can use a simpler rule but just as a quick review this is of course the power rule right over here very handy for taking derivatives of X raised to some power it's also we can use that with the derivative properties of sums of derivatives or drift or differences of derivatives to take derivatives of polynomials this right over here is the product rule if I have an expression I want to take the derivative of and I can think of it as the product of two functions well then the derivative is going to be the derivative of the first function times the second function plus the first function times the derivative of the second once again if this looks completely unfamiliar to you or it's a little your little shaky go watch the videos do the practice on the power rule or the product rule or in this case the quotient rule and the quotient rule is a little bit more involved and then we have practice and videos on that and I always have mixed feelings about it because if you don't remember the quotient rule you can usually or you can always convert a quotient into a product by expressing this thing at the bottom as f of X or by expressing this as f of X times G of X to the negative one so you could take the derivative with a combination of the products and this fourth rule over here the chain rule and if any of this if any of this is looking unfamiliar again don't watch this video this video is for folks who are familiar with each of these derivative rules or derivative techniques and now I want to think about what what our strategies for deciding when to apply which so let's do that let's say that I have the expression let's say I'm interested in taking the derivative of x squared plus X minus 2 over X minus 1 which of these rules or techniques would you use well you might immediately say hey look this is look like a rational expression I could say look I could say this is my f of X right over here I could say this is my G of X right over here and I could apply the quotient rule this looks like a quotient of two expressions and you could do that and if you do all the mathematics correctly you will get the correct answer but in this case it's good to just take a little time to realize well can I simplify this algebraically so maybe I can do a little bit less work and if you look at it that way you might realize like what if I factored this numerator I can factor it as X plus 2 times X minus 1 and then I could cancel these two characters out and say hey you know what this is going to be the same thing as the derivative with respect to X of X plus 2 derivative with respect to X of X plus 2 which is much much much much much more straightforward than trying to apply the quotient rule here you just take the derivative with respect to X of X which is just going to be 1 and the derivative with respect to X of 2 is just going to be 0 and so all of this is just going to simplify to 1 for taking the derivative of that you're essentially just using the power rule and so once again just a simple algebraic recognition things become much much much much more simple let's do another example so let's say that you you were to see or someone were to ask you to take the derivative with respect to X of let me see so let's say you had x squared plus 2x minus 5 over over X so once again you might be tempted to use the quotient rule this looks like the quotient of two expressions but then you might realize well there's some algebraic manipulations I can do to make this simpler you could express this as a product you could say that this is the same thing as and I'm just going to focus on what's inside the parenthesis or inside the brackets this is the same thing as X to the negative 1 times x squared plus 2x minus 5 and then you might want to apply the product rule but there's even a better simplification here you could just divide each of these terms by X or one way to think about it distribute this 1 over X across all the terms x to the negative 1 is the same thing as 1 over X and if you do that x squared divided by X is going to be X to X divided by X is going to be 2 and then negative 5 divided by X well you could write that as negative 5 over X or negative 5 X to the negative 1 and now taking the derivative of this with respect to X is much easier than using either the quotient or the power rule this is going to be let's use derivative of that it's just going to be one derivative of 2 is just going to be 0 and here even though you have a negative exponent it might look a little intimidating this is just take it using the power rule so negative 1 times negative 5 is positive 5x to the if we take one less than negative 1 we're going to go to the negative 2 power so once again making this algebraic recognition simplified things a good bit let's do a few more examples of just starting to recognize when we might be able to simplify things to do things a little bit easier so let's say that someone said hey you take take the derivative with respect to X and I'm using X as our variable that we're taking the derivative with respect to but obviously this works for for any variables that we are using so let's say we're saying square root of x over x squared pause this video and think about how would you approach this if you want to take the derivative with respect to X of the square root of x over x squared well once again you might say this is a quotient of two expressions might try to apply the quotient rule or you might recognize well look this is this is the same thing let me just focus on what's inside the brackets you could view this as X to the negative 2 times the square root of x times the square root of x and then you might want to use a product rule but you could simplify this even better you could say this is the same thing as X to the neck two times X to the 1/2 power and I just using our exponent properties negative 2 plus 1/2 is negative 3 halves so this is the derivative of X to the negative 3 halves power and so here once again we took something that we thought we might have to use a quotient rule or use from the pot use the product rule and now this just becomes a straightforward using the power rule so this is just going to be equal to so bring the negative three-halves out front negative 3 halves X to the negative 3 halves minus 1 is negative 5 halves power so once again just before you just especially if you're about to apply the quotient rule and sometimes even the product rule just see is there is there an algebraic simplification sometimes the trigonometric simplification that you can make that that eases your job it makes things less hairy and as a general tip I can't say this is going to be always true but if you're taking some type of an exam and you're going down some really hairy route which the quotient rule will often take you it's a good sign that he take a pause before trying to run through all of that algebra to apply the quotient rule and see if you can simplify things so let's give another example and this one there's not an there's not a must there's not an obvious way and it really depends on what focuses preferences are but let's say you want to take the derivative with respect to X of 1 over 2x to the negative 5 sorry 1 over 2x minus 5 I should say well here you could immediately you could apply the quotient rule here or the numerator you view you view that as f of X you could view this is the same thing as the derivative with respect to X instead of 2x minus 5 we do that in the blue color 2x minus 5 to the negative 1 power and now this in this situation you would use a combination of the power rule and the chain rule you'd say okay my G of X is 2x minus 5 and F of G of X is going to be this whole expression and so if you did if you applied the chain rule this is going to be the derivative of the outside function our f of X with respect to the inside function so or the derivative F of G of X with respect to G of X so it's going to be negative we'll bring that negative out front so essentially just going to use the power rule here negative 2x minus 5 to the negative 2 and then we multiply that times the derivative of x the derivative of the inside function so the inside functions derivative the derivative 2x is 2 derivative negative 5 is 0 so it's going to be times 2 and of course you can simplify it so it's negative 2 times all of this business let me do one more example here just to hit the point home and once again there isn't a must way there's a way that you have to do this but just let you appreciate that there's multiple ways to approach these types of derivatives so let's say someone said take the derivative of 2x plus 1 squared pause the video and think about how you would do that well one way to do it is just to apply the chain rule just like we just did so you could say all right it's going to be the derivative of the outside with respect to the inside so it's going to be 2 times 2x plus 1 to the first power taking 1 less than that times the derivative of the inside which is just going to be 2 and so this is going to be equal to 4 times 2x plus 1 which is equal to week if we want to distribute the 4 we could say it's 8x plus 4 that's a completely legitimate way of doing it now there are other ways of doing it you could expand out to X plus 1 squared you could say this is the same thing as the derivative with respect to X of 2x squared is going to be 4x squared and then 2 times the product of these terms is going to be plus 4x plus 1 and now you would just apply the power rule so a little bit of extra algebra up front but you can just go straight forward with the power rule and you're going to get this exact same thing so the whole takeaway here is pause look at your expression see if there's a way to simplify it and it's especially a good thing if you can get out of using the quotient rule because that sometimes it's just hard to know or remember and even when you do remember it can get quite hairy quite fast