Manipulating functions before differentiation

- [Instructor] 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 to 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 this is 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 differences of derivatives to take derivatives of polynomials. This right over here is the product rule. If I have an expression that 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 you're a little shaky, go watch the videos, do the practice on the power rule and the product rule, or in this case the quotient rule. And the quotient rule is a little bit more involved and we have practicing 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 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 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 want to think about well what are 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 two over X minus one. Which of these rules or techniques would you use? Well you might immediately say hey look this looks like a rational expression, 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 wait what if I factored this numerator I can factor it as X plus two times X minus one. And then I could cancel these two characters out and I can say hey you know what this is going to be the same thing as the derivative with respect to X of X plus two. Derivative with respect to X of X plus two, which is much much much much more straightforward than trying to apply the quotient rule. Here you would just take the derivative with respect to X of X which is just going to be one and the derivative with respect to X of two is just going to be zero and so all of this is going to simplify to one. If we're 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 more simple. Let's do another example. So let's say that 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 two X minus five 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 there's some algebraic manipulations I could 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 parentheses or inside the brackets, this is the same thing as X to the negative one times X squared plus 2x minus five 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 one over X across all the terms, X to the negative one is the same thing as one over X and if you do that X squared divided by X is going to be X. 2x divided by X is going to be two and then negative five divided by X, well you could write that as negative five over X or negative five X to the negative one. And now I'm 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 see the derivative of that is going to be one, derivative of two is going to be zero and here even though you have a negative exponent, it might look a little intimidating, this is just taken using the power rule. So negative one times negative five is positive five X to the, if we take one less than negative one we're going to go the negative two 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 the derivative with respect to X and I'm using X as our variable that we're taking the derivative of with respect to but obviously this works 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 the same thing, let me just focus on what's inside the brackets, you could view this as X to the negative two times the square root of X and then you might want to use the product rule but you could simplify this even better. You could say this is the same thing as X to the negative two times X to the one half power and now just using our exponent properties, negative two plus one half is negative three halves, so this is the derivative X to the negative three halves power. And so here once again we took something that we thought we might have to use the quotient rule or 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 three halves, X to the negative three halves minus one is negative five halves power. So once again just before you, especially if you're about to apply the quotient rule and sometimes even the product rule, just see is there an algebraic simplification, sometimes a trigonometric simplification that you can make that eases your job that makes things less hairy? As a general tip I can't say this is going to be always true but if you're taking some type of exam and you're going down some really hairy route which the quotient rule will often take you, it's a good sign that hey 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. In this one there's not an obvious way and it really depends on what folks' preferences are, but let's say you want to take the derivative with respect to X of one over 2x to the negative five, sorry, one over 2x minus five I should say. Well here you could immediately apply the quotient rule here the numerator you view that as F of X. You could view this as the same thing as the derivative with respect to X. Instead of 2x minus five, let me do that in the blue color. 2x minus five to the negative one power. In this situation, you would use a combination of the power rule and the chain rule. You would say okay my G of X is 2x minus five and F of G of X is going to be this whole expression. And so 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. The derivative of 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 we're essentially just going to use the power rule here. Negative 2x minus five to the negative two and then we multiply that times the derivative of the inside function. So the inside function's derivative, the derivative of 2x is two, the derivative of negative five is zero so it's going to be times two and of course you can simplify so it's a negative two 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 isn't 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 one 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 alright here's going to be the derivative of the outside with respect to the inside. So it's going to be two times 2x plus one to the first power, taking one less than that times the derivative of the inside which is just going to be two and so this is going to be equal to four times 2x plus one, which is equal to, if we want to distribute the four, we could say it's 8x plus four. That's a completely legitimate way of doing it. Now there are other ways of doing it. You could expand out to X plus one squared. You could say hey this is the same thing as the derivative with respect to X of 2x squared is going to be 4x squared and then two times the product of these terms is going to be plus 4x plus one. And now you would just apply the power rule. It's a little bit of extra algebra up front but you can just go straightforward 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 'cause that sometimes is just hard to know or remember and even when you do remember it, it can get quite hairy quite fast.