all right so today we're going to talk just a little bit about implicit differentiation but business really just the things which you already know right okay so I'm just going to work a couple of more examples of it and then we're going to try to see a pattern in the way it works so that we can write down a quicker way to do it which is alluded to this last week so this will not take up the whole time and afterwards we can talk about anything you want to and you have questions you have with the midterm coming out let's start with an example let's say we're going to do two functions a 3x squared Y minus sine of X y plus y to the one-half equals let's say X to the 3 minus tasks up there I'll just say execute nothing too nasty and the question is find dy/dx okay which if we want to make things look a little simpler we can always write dy DX as Y prime Y is a function of X so Y prime just means the derivative of Y with respect X so to set up the nice trick for doing this I want us to get into a hat okay here's the hat move everything one side okay so here you have an equation something equals something else and I want to take everything and move it to one side to it I'm left to something equaling zero okay at first this isn't going to make a whole lot of sense to why we're doing this but later on you'll see this is going to set things up quite nice so if we do that we can rewrite this as 3x squared y minus sine of x times y plus y to the one half and then we have to subtract X cubed and y fue and we get something equaling zero now if we eventually want to compute a derivative at some point we have to take a derivative okay now seems like a good time we have some long function I really it's really what's called an implicitly defined function all right so implicitly defined functions or when you're going to have powers of Y or functions of Y side also and it equals zero okay so whenever this expression is equal to zero so when we take a derivative the derivative will also equal zero yeah sure you have a function equal to zero and take the derivative it's still zero okay fine so let's take a derivative well how do we attack this I'd use the product rule right because I have two functions x squared and y okay so the product rule says if the three just kind of sits on the outside I do the derivative of x squared which is x times the other function one plus and now I fix the x squared and I take the derivative of Y what's the derivative of y dy DX or if you like Y prime okay fine just put Wi-Fi okay the next point I have to do a - okay now again I have a product of functions so I have to use the product rule the derivative of sine is cosine times y plus and I fix the sign and I have to take the derivative of Y which we already saw is Y prime I go to the next one I have Y to the one half what rule do I have to use to differentiate this power rule and the chain rule right is y is itself a function of X right I have a function of a function or a dive right this way a function of a function so the power rule says I'll get one-half Y to the negative one-half and the chain rule says I don't have to multiply by Y prime alright chain rule says you do the derivative of the outside times the derivative of the inside next comes the minus X cubed what's the derivative that minus 3x squared and then comes minus y cubed would I use for that minus 3y squared times the derivative of Y right use the chain rule again and this whole expression has equal zero it's a long expression can we zoom in the target assumed it's so long okay now I want to find y prime I'm looking I have these thanks why Prime's all over the place and I want to isolate just that lag crime so this is just algebra now we're solving for y prime okay so here's the trick everything that doesn't have a wide Prime in it we're going to throw on to the other side okay now what are all the things that don't have a y Prime in them well I can look right I can visually see this thing right well this one this one don't have Y Prime and those are exactly the ones where I took a derivative of the X part of the term and I didn't take the derivative of the Y part of the term okay so for instance here I use the product rule and in the first case I just differentiated the X part and in the second part I differentiated the Y and that's how I got a Y prime same thing happen here when I differentiated just the X part of the function I didn't get a wife I'm gonna like differentiate it the y part of the function I did okay here there's no X here I got an X and there was no y Prime in here there was no X in you get a Y prime okay so keep that in mind okay so I'm gonna take everything without a Y Prime and throw it to the other side so that's going to give me and I do make sure I expand all these there's a three there's one minus one and so forth so let's say I get 3x squared Y Prime and then I'm going to throw this away I get minus one thing I left it correctly up I get minus sine of X Y Prime and then I get here a plus 1/2 Y to the minus 1/2 times y prime well I had a minus here so it became a minus sign times y prime alright so remember I had - this expression when I took the derivative all right both terms of the derivative coming from the product rule we're going to get typical - okay that's going to X oughta move over and here on the month of the - 3 y square like right and this is going to equal well now I throw all the X's over so here was a 6 XY so it becomes minus 6 X Y okay that was Y prime if you hear I get a minus cosine of X so become a plus cosine of X times y and then Y prime Y prime okay this is a - trick square I move it over to the plus okay now each term on the left has a y Prime that's good I'm trying to solve for y prime so i factor out the Y prime and I get Y prime times 3x squared minus sine of X plus 1/2 Y minus 1/2 minus 3 y squared equals this outer side minus 6x y cosine of X y plus 4x squared and how do I now solve for y prime I just divide right by everything next to the Y Prime and so dy/dx or Y prime is equal to negative 6x y plus the cosine of x times y plus 3x squared divided by now all this junk 3x squared I think minus sine X plus 1/2 Y to the minus 1/2 minus 3y okay so this is pretty nasty right and look at all the work we had to do and look at all the places we could have made an arithmetic mistake or an algebra mistake okay so this is I mean it works but somehow it's not optimal for test taking conditions right who in here thinks they would get you know a minus sign mixed up in the process on a test exactly okay luckily there's a way to do this all once okay so you just write down the eight you have to work through all the rhythm and the way to do it is to just take the process in your mind and do it all at once okay so let's figure out what I mean by we implicitly differentiate things we do the derivative we subtract the x's to the other side and then we divide by the Y's okay so the X is were the terms where we actually differentiated the X part and the Y's or where we differentiated the y part now if I remember right there's a I'm using a band called oasis or if you guys heard of them okay so they did a song about this in fact ladies I think that it went something like even place simply did the Ranchi a gated move the X's over by subtracting then divided the wise to put them on the other side do something like that I think okay so yes so problem doing it again yeah listen we did the NGA he added subtract the x's over and divide to get the Y's on the other side everybody implicitly did renji a he ate it subtract T X's over then divide to get the Y's on the other side come on alright so it's the same thing every time you do this right implicitly differentiate subtract the X is / / wise to get them on the other side okay so what does that mean it means that the derivative dy/dx is given by Stu region it was like a constant when I did it here right and I took the derivative the X part you got cosine and then the line is stuck around just like if it was a 2 when I use this symbol they oftentimes pronounced del I'm saying take the derivative but anytime you see a variable like a Y or a U or W or Z or anything which is not an X pretend it's a constant okay this is called a partial derivative this is really a subject or calculus 3 multivariable calculus ok but we didn't without introducing it formally we can just use it okay so what do we do we subtracted the X derivative and then we divide it by the Y derivatives so we just divide now by the partial derivative with respect to Y you mean you treat the X like a constant and look what happened right when we did the second part of the product rule on this term and the x squared just stuck around because we differentiated the y part need to fix the x squared mark same thing here all right the sine of X just a constant okay we just treated it like a constant okay the coursers no other X's anywhere problem okay so let's write down the answer just by using this formula so we have to put down a - very certain life and now we're going to differentiate with respect to X meaning we treat everything not an X as a constant so we take this function we say okay if we're doing with respect to X Y is just a constant okay so it's just you can pull it out it's like 3y that's your constant times derivative of x squared okay so what's the derivative of 3x squared Y it's going to be exactly right you put the Y in the front use okay now I have 6x times y so I just go down and 6 that's why okay now I go to the next term I treat the line just as a constant it's like a - hey what's the derivative of sine of X times a constant exactly right cosine of X times a constant right so it could be cosine of x times 1 minus cosine of X times y is next one Y to the one half we're differentiating with respect to X which means Y is a constant so what's the derivative zero right this is y to the 1 half that's just a constant to the one half all right still a constant to the derivative the zero we don't worry about it next term minus X cubed what's the derivative minus 3x squared write it down last term Oh 2y cubed all right it's just a constant again so the derivative is zero we don't have to worry about / now when you do the whole derivative again but this time we treat X as a constant okay y is not a variable what's the derivative of 3x squared Y remember Y is the Y is now our variable right so it's just like we're differentiating with respect to X right driven Y would be 3x yeah if you have the 3 x squared right it's just the content you have a constant times y the derivative with respect to Y is just the constant 3x squared that's a 3x squared next one alright if this was 3y squared X and I said what's the derivative if Y was constant you'd say well it's something times X so it's just something right now I'm doing it with respect to Y okay so I'm on this bottom half or a difference in step two Y so now Y is my variable and X is the constant that makes our okay so I just have 3x squared times y so the derivative is 3x squared okay in the next one I have sine of X times y and I'm differentiating with respect to Y so what's the derivative it's just sine of X I put minus sine of X and next one is y to the one-half just use the power rule 1/2 Y and then the last okay then I have an X what's the derivative xq0 all right they're doing it with respect to Y then the last one is y2 and that becomes 3y squared minus 3y squared now look at the answer that we got and look at the one we up before the only difference right is here they're combining outside and here there's - up top but I can fix that immediately by making out a minus plus plus by distributing okay so you can see that the value of this method is you're taking all the work you're doing and you're abstracting it I'm saying we would hold on a second I keep doing the same process every time is there some way to just do it all at once yeah there is okay you differentiate training why is it constant and then differentiate treating X as a country okay now a word of caution this works when you have two variables x and y but there's nothing to stop me from coming up with some sort of relation where I have a third variable right I can have a you of these W okay and then of course this formula doesn't work so nicely and because you don't just have me feeling so you'd have to do that you have to work that out it's done this is just really slick let's do it I got a not answer the question how would you do that though or if you want two variables except one is like flat iron that's an excellent question let's look at okay let's say we let's do something very simple okay so here y and z our function FX so I could ask a lot of different questions I could ask what is dy/dx I could ask what is DZ DX if I really was mean I could ask what's dy DZ or DZ dy I can ask these questions is how do you have to attack these things well you have to use the chain rule again right and of course here you have to use the product rule to two packages so for instance if we take the derivative of X Y Z that's three functions multiplied together you remember the rule when you had more than two product in a product right you fix two of them and differentiate the third then fix a different one or fix it different to differentiate the third and fix a different to differentiate thirteen three terms okay so the first one we'll say we'll fix y&z and you differentiate X and we'll do everything with respect to X here right so derivative X is 1 so you get y z+ okay now we'll fix X and Z all right and then multiply by the derivative and by the derivative of Y you get X Z Y prime it now we'll fix X and why take the derivative of Z the Z Prime and this is all equal to zero and if we're looking for dy/dx then we have to solve for y Prime well we just put everything not a Y Prime on one side everything with high crime on the other side divide by the coefficient in front of the Y Prime and you get Y prime equals minus y Z minus XY prime divided by XP and without knowing what x or y and z are for she can't say anymore but this certainly worked right this is dy/dx you can see that the change of Y with respect to X actually depends upon Z but I mean this is what starts happening when you allow a lot more variables into your equation or and things get more complicated ultimately more rewarding more realistic because it turns out in the real world single variable calculus is just not good enough usually things have more than one input right more than one independent variable to worry about and are often more than one I'll do another example just abusing this method if you much faster than the first one because now can just use the formula and you see it doesn't matter I mean I can make it as complicated as I like it's not so hard let's say you have 2x e to the Y plus sign x squared plus y times cosine Y minus 3x ok so I've already hidden in here that I'm putting everything on one side as I did that step for you and we want to find dy/dx okay so I can use my formula first part of formulas very easy I write down - I remember where the minus sign came from subtracting over the derivatives of the x's of course that's the next thing we put down on the derivatives of the XS okay so let's differentiate differentiate this where what's a constant why is it constant right so what about you the why it's a constant right so when I differentiate this what do I get - e to the Y DX is my variable right the derivative X is just 1 like those always get to e to the Y okay now I'm going to do this one okay now let's see I'm doing goats respective X so this Y is a constant so it's like x squared plus 1 so what rule do i fuse on this chain so I get 1 okay the derivative of the inside or the outside is derivative of sine which is I'm sorry I just leave the inside the sine T times the derivative of the inside to it which is 2x notice this cosine of Y doesn't hurt me because it was just a constant so I just couldn't leave it I didn't have to use the product and I get to this last step when we have derivative of minus 3x which is going to be minus okay now I go back and I do the whole derivative again but this time why is my variable and X is my concert okay so X is a constant so this 2 X's could be left what's the you are going either Y e to the Y so I much of the same thing 2 X to the 1 plus okay now I will have to use the product rule all right because I have two functions of Y okay so first I have to differentiate say this one I get cosine of x squared plus y times the derivative of the inside what's the derivative now x squared plus y it's just one right the x squared is constant that goes away Uruguay is one it times cosine of Y and plus okay now I need to do the derivative or I can fix the hat and I can do the derivative of cosine of Y which is minus something okay so that means I'm going to actually go - okay and then the last term is a minus 3x and the derivative of that will be language tips you know actually be 0 because X we're treating is a constant group tough okay there's your answer and we we did no work you know that I mean you can't even show work when you do it this way or you just write down the answer okay the old calculus teacher - hey yeah we're here okay on the top I imagine you mean because now this is a constant all right why is just a constant when we differentiate with respect to X so it's just like a - so we just throw it in at the end like if you choose to use this method on your midterm exam which is quite okay write down this formula with it so I know that's what you're doing otherwise I have no idea if you're just guessing or if you have anything okay you can still make a mistake doing this okay and if you make a mistake and now there's no work how can I give you partial credit no okay so the least write down the formula you're using okay very good any questions go get comes already if there's like no lies easy easy two variables yeah okay so let's think about this formula here and you're supposed to subtract the derivatives of the X's and then divide by the derivatives of the whites but if there's no derivatives of the Y's then then you just get what well you should actually maybe the gut instinct to say well zero because it's none but it's actually one right you just have one Y Prime on one side so if I write Y as some function okay well the first rule license put everything on one side okay so you might sales by nine y minus f of X zero now when I differentiate I get Y prime minus F prime before zero okay and then of course I put all the things without of Y Prime on one side and I just get Y prime equals F prime of X okay which is of course from here that's what you get exactly it's not surprising that is such a neat thing to say okay so there's no there's really no advantage to it because of all you'd be saying is well this formula be telling you okay how would you how would you do it from here to use this formula alright it's okay first go through and differentiate okay perfect you get dy/dx equals minus and you do the derivative with respect to X so the Y goes away and you get minus F prime of X then you divide by the derivative of Y which is like one with respect to Y and course now you have - - and you get a plus so you haven't you haven't helped yourself any so tell me you can't use it it says it's overly complicated yeah it's like beating a snake of your garden hose okay so any questions about this is that the wrong expression or is it supposed to be beating us your garden hose with the snakes never heard that really interesting okay any questions about this okay let's go so ah so feel free to use this method right I think it's shorter now the other good Google is also never I love this I can make up nonsensical expressions and they can be shot down as being bunk within moments what a time to be alive I know that's not true you guys haven't seen the pictures of me I'm not those kind of pictures the other kind of pictures that's probably not that you'd probably means you think the wrong thing no not that kind of pictures the under kind of pictures not figuring out how to say this problem okay so at this point I am NOT going to introduce anything new before so you now know everything that you need to know what mr. I'm not going to throw any more curveballs at you it's going to be straight forward do you know this stuff I can throw curve balls at you but I'm more of a fastball pitcher nice I did what did you want one too what would have been nice if you showed up on time huh that's the trade show everybody has an excuse that's why I take the 8 o'clock train every morning because I can't be late for you guys ok so if you want to ask questions now's a good time right you want to find out what's going to be on the midterm well I told you blessed few weeks but if you I mean you want ask questions substantial questions about how to do problems anything you want please exit it'll be good we'll have some good work problems for the video a lot of you asked me for example so you have anything you want to see how before trying to find started like a union service with some question good question good question what's that so that's actor yeah okay so so for instance if I give you a function like this then what's the derivative what's going to be a problem perhaps zero right now we haven't proved that this function doesn't have a derivative it's there if you drop the sign then we know for sure that it doesn't have a derivative at zero with the side who knows maybe it you know somehow smoothes it out a little bit it turns out it doesn't turned out that the sine function is going to be just as bad with an absolute value on the inside is it's just absolute value less so you know you you always want to look at a function say does this make sense am I always going to be able to find a derivative if I'm asking you to evaluate the derivative at a point does the function even exist at that point of course this is something good to always check another reason why you want to check is why I always believe think when you see a problem right before you actually just start doing it think abut and for instance you remember in this homework problem where we had we have that last homework sheet with yeah right you have on his right something I'm right you might have something like sine of secant inverse of cotangent of e to the ln of cotangent inverse secant of arc sine X all right and I say fine derivatives huh my missing son or one five six seven four you know an exponential notation from multiplying two in fact I need parenthetical notation for many parentheses parenthetical notation how would you find the derivative of this alright late yell out the answer on why that's really quickly if I can figure it out like did you figure that out - okay okay so so you see if you look at this problem right oh and this looks hideous absolutely giddy that's right then if you kind of peek around at you go wait a second there's me in LA all right those are inverse functions so they're going to cancel out this is the same as the derivative of sine of secant inverse of cotangent Oh cotangent inverse of secant of arc sine of X right and then from here you all the way to second now I see what's going on the cotangent and the cotangent inverse er and they're they cancel out then I have a secant inverse of a secant ha ha ha ha and then I have a sonnet an arc sine I'm just left with the derivative of X so this is one where it pays to look at the function first to look for this I'm not saying this problem beyond the text of the same you always look to function and make sure everything looks right that it's not going to automatically simplify hey I make mistakes sometimes but I found a problem just yesterday's it is a piecewise defined function that's on the midterm and there was a mistake in it happens it was your fault so this is you know I don't make mistakes with my school absolutely everything I do is absolutely appreciated all right more questions and you would have swear to change only top x the reciprocal of our markers they're gonna help with the way we found out how to do or maybe that was you know when you will put my conch in all right so let's try I have to make one up so can only hope that it works we'll try to limit as X goes to 2 of square root X plus 2 minus 2 and on top we'll have x squared minus X minus 2 I just made this up on the spots that was my failed miserably but let's see what happened if you try to plug a 2 into the bottom right you can get the square root of 4 which is 2 minus 2 which is 0 okay so this bunk the value 2 is not in the domain you can't just plug it in so Stephanie's question suggests the intended solution which is to multiply top and bottom by the conjugate the denominator we'll see it just does anything nice might not but at least the method right you see you would immediately have to do okay so on top ooh we get nasty let's do the bottom that's easy alright on the bottom we get x + 2 - cool hmm law school - cool that's right so we're going to get an X plus 2 from the first term right as we do these two squared becomes X plus 2 and then we're going in - oh that's not good all coming crashing down so we already see what we're gonna be intro that's going to end up being X minus 2 seems already be worse but okay we'll figure out the top maybe something nice to happen to little cystic all right so you get x squared times square root of x plus 2 and we get 2x squared and then we have minus x times square root of x plus 2 and we have minus 2x and we have minus 2 square root of x plus 2 and we have minus 4 Oh okay I'm not encouraged yep cyclic so x squared minus X minus 2x squared we love the limited concept with the humanity so let's see this is the need and minus two times one but do I so do I have that dominate so is X minus 2 y am and I don't have this in the topic well I sorta do I have 2x squared minus 2x minus 4 so okay let's rewrite this I'll show you we're still have a problem the top becomes alright let me just actually just write all the things in front of X square root of x plus 2 maybe actually yeah maybe this works out nicely X then get x squared minus X minus 2 times square root of x plus 2 all right those are the first third fifth term through the other one you get a 2x squared a minus 2x and minus for each of those has a 2 which I can factor out so we get a plus 2 times x squared minus X minus 2 now I'm asking why in the world that I even expand any of this if I just left it like this in the first place I wouldn't have any problems we actually could have factors right away but we don't know that right we just worked it naively okay so if we factor out this person I mean these two terms right exactly back to multiplying this times this okay fine so we'll just rewrite it like that sometimes you do the right thing and you get back where you started that's okay so you have x squared minus X minus 2 times the square root of x plus 2 plus 2 over X minus 2 and now if you say pointed out this factors as X minus 2 times X plus 1 X - 1 x squared - 2 over X minus 2 all right and now the X minus 2's will cancel and we have X plus 1 times the square root of x plus 2 and now there is no domain problem all right these are all continuous functions so I can just plug into it if I get three times okay that's a 4 square root of 2 plus 2 is 4 12 Q a loved one had hoc examples workout correct okay but we could have actually saved all these steps if we have just multiplied and then went right to this we have two right here all this stuff with garbage but you don't know that everything I read this is the problem with mathematics right this is how we worked it right we did it very naively but that's how else can you do it right okay so you go through and you do it naively and then you figure out oh wait we could have just went from right here here and then when you write it up and you publish a paper you just write this there for anything so how do you know not to multiply it out that's because you did it work - completely confusing everything ok let's erase we started with just this we tried to immediately plug into and we saw it we can't do that zero off the bottom yeah so we said okay fine there's a standard trick whenever you are in these limit situations and you see on Newton a denominator which is going to zero and you know something minus something right specially when there's something over here is a square root it seems but it doesn't need to be then the trick seems to be to multiply by the conjugate top and by yep happy about that everybody happy about that okay fine from here well if you multiply the denominators you get X minus 2 okay that's not so tricky right which is utilizing the whole a minus B times a plus B equals a squared minus B squared difference of squares formula fine on top well if we just don't do anything except write them next to each other then we're actually much better shape than if we expand it up why are we in much better shape and if we expand it out well turns out that expanding it out just lead you back to putting it back into the school is it close we have the negative just before okay so the square is X plus two squared minus two squared X plus 2 square root squared is X plus 2 two Spurs for I plus 2 minus weights but problems like this to tell it this nice I don't write problems that come out ugly even on the homework did you notice every single problem just beautiful in the end amazing alright and not putting anything on there that's supposed to test your ability to massage equations and you know be an accountant I want to know if you can find limits take derivatives when years the thing in the real world these are things they're usually pretty easy or they're so complicated that you just need to know in theory what's going on and how to plug it into a computer algebra system alright like a maple or Mathematica or something you plug it in and go build computing and some nasty derivatives or limits or something that's this is one of the problems I problems are either completely trivial or they're beyond what you really have the time to do so I'm not testing your ability to keep track of numbers running across the page and there is no much time it will depends on you know how fast your high I mean I did your midterm last night it took me seven minutes that gives me lots of time alright so what was this the only issue you had Stephanie was the X minus two okay so in from there right we're just factor I'd love to show you I just unsolved that question well this one no two though I lost time I did different for it but oh yeah I mean there's the standard way that we do this right the one that's part using a chain rule where you've got Y Prime's popping up everywhere yeah okay so that's what you missed all the first half hours I gave an alternative approach which utilizes the methodology of the first approach but does it all at once both both ways are acceptable ways okay okay so this is why you come late yeah get out things word problem is it word or worm way okay good good yeah there are no I could make a word problem about worms if you like but why'd you bring up worms I didn't a that word hey yeah word problem that neither there's nothing right to me the word problems are part of the applied part of the course which is what comes after the altar so now these are I'm really focusing on can you take a limit can you find the derivative and you know what these things mean okay so when I said that the homework problems are the best represent representatives for midterm problems not lying to you what's the some of them at all though I guess there was one there's like a 1 on the circle yard okay that's nothing there's nothing on there that's going to make you think you know like oh wow that's really cool is nothing like that on this test okay that's for the second half of the course there's one problem on it well 3 X 3 bar that asks you to explain certain concepts so that's probably the only part of the step size what's that angle yeah it's actually able to explain this concept and when I write explain this concept I don't mean you know give me the precise mathematical definition unless you want to that's fine but you don't have to you can use just ha go go on what you want more I could put more problems that require either you want worms too I don't know you guys have a fascination with worms by the way as usual look out for red herrings they are on the test we've talked about red herrings before I don't think we have here we're talking about director Colin that is what's that where the actor is I look it up Kalman or a very like something you think of something else is that what Google sensors I'm used to that's exactly the definition that's cell phone first you come in late and you don't turn the cell phone off cool it you didn't come late all right wear a helmet our little Johnny was a half an hour early I guess literary narrative element intends to strike the rear for more important than quality plate equipment the term red herring originates with the tradition whereby young hunting dogs in Britain we're trained to follow eccentric views of red herring this mangy fish should be dry go on the trails and popular and follow the scent later when the dog later when the dog was being trained follow the faint odor of a pond so badger the trainer would drag a red herring to the much stronger order across the animal trail right angles God would eventually learn how to follow the original you guys get the idea of this you remember we had a problem and I believe we even use this expression to describe it when I had to draft I don't remember the exact problem and it looks something like this this is simplified version of it and graph the absolute value of sine of X plus 4 now what is sine of X look like well maybe that's my sing going up down between 1 minus 1 when you add 4 just the grass 1.4 right so now it's going between 3 & 5 ok so when you take the absolute value what happens that's it right this is between 3 & 5 so those absolute value signs are a red herring they're there to make the problem look harder than it is maybe if they go there's something weird going on but of course this is equal okay okay so that's a red herring something which is just there to distract you and make you think oh my gosh this is hard oh my goodness how am I going to deal with that but then it turns out has nothing to do with the problem it's very easy and I can already I can think of at least one red herring on the exam so if you go to look up my two room it doesn't mean they're right having us on correct at all let me know some curriculum read the definition a red herring would more be like read the definition again would be like a year as a problem either we have to do it one way we have to do no no no no it's not a red herring that's misdirection that's not that's the Joe Gibbs is running game now reread the definition again that's like the counter try right Campinas that were wrong exactly Oh read it read again read not about the whole thing about the dogs that is kind of a red herring but not really you know I think when I get older I will start flying chocolate students but thank you guys all right you guys well I don't like all of you there's at least one of you in here I don't like anybody want to guess let's take a vote do do I not like here it's just one person at least edit not you there is still one ID did you not find it yet at the end of the Internet well not indeed I was right and you were just we'll leave it at that everything up the graph things Brad was like well you know that's our glaze and find a brat and I'm good crack yourself that's a good idea no cheese that's a bet oh my god make a note to myself after that and then they count is your submitted problem right back in after worry well no when you have that problem on the exam you know I'm going to put on the test Stephanie's problems I can thank her this one is worth most of the points you graph it incorrectly but you're not the one what did you find it Stiller I feel like other math classes have to worry about like the language barrier and I just don't understand half the metaphors that you can five years I have to tell you half the metaphors that I use I make up on the spot and half of those I intend to be complete nonsense loses in the red area right the actual box yeah so like the Fox things that I feel that way and find that box but actually this one and this problem for instance the absolute values make you think the important part of this rights you know which side of X plus four is the important part oh my guns we gotta figure out what's happening the absolute value of x totally are you said it'll be harder objection sustained the castles question occur not a variable calculus Kings definitely should we do some more calculus final video I'm in it yeah let's listen if you have no questions we don't I mean I'm not going to answer any you have