• [categories_slider autoplay="1"]
  • Lec 1 | MIT 18.01 Single Variable Calculus, Fall 2007

    Copy Help
    • Public/Private: Change the visibility of this video on your My Videos tab
    • Save/Unsave: Save/Unsave this video to/from your Saved Videos tab
    • Copy: Copy this video link to your system clipboard
    • Email: Copy this video link to your default email application
    • Remove: Remove this video from your My Videos or Saved Videos tab
    search-icon
    Watch at: 00:00 / 00:00:20the following content is provided undera Creative Commons license your supportwill help MIT OpenCourseWare continue tooffer high quality educational resourcesfor freeto make a donation or to view additionalmaterials from hundreds of MIT coursesvisit MIT opencourseware at ocw.mit.eduWatch at: 00:20 / 00:40so again welcome to 1801 we're gettingstarted today with what we're callingunit 1 highly imaginative topic highlyimaginative title and it'sdifferentiation so let me first tell youWatch at: 00:40 / 01:00briefly what's in store in the nextcouple of weeks the main topic today iswhat is a derivative and we're going toWatch at: 01:00 / 01:20look at this from several differentpoints of view and the first one is ageometric interpretation and that's whatwe'll spend most of today on and thenWatch at: 01:20 / 01:40we'll also talk about a physicalinterpretation of what a derivative isand thenWatch at: 01:40 / 02:00there's going to be something else whichI guess is maybe the reason why calculusis so fundamental and why we alwaysstart with it at in most science andengineering schools which is theimportance of derivatives of this to allWatch at: 02:00 / 02:20measurements so that means pretty muchevery place that means in science andengineering in economics in politicalscience etc polling lots of commercialWatch at: 02:20 / 02:40applications just just about everythingnow so that's what we'll be gettingstarted with and then there's anotherthing that we're going to do in thisunit which is we're going to explain howWatch at: 02:40 / 03:00to differentiate anything so how todifferentiate any function you know andWatch at: 03:00 / 03:20that's kind of a tall order but let mejust give you an example if you want totake the derivative this we'll see todayis the notation for the derivative ofsomething of some messy function like eto the X arc tan of X we'll work thisWatch at: 03:20 / 03:40outby the end of this unit all rightso anything you can think of anythingyou can write down we can differentiateit all right so that's what we're goingto do and today as I said we're going tospend most of our time on this geometricWatch at: 03:40 / 04:00interpretation so let's let's begin withthatso here we go with the geometricinterpretation of derivatives and whatWatch at: 04:00 / 04:20we're going to do is just ask thegeometric problem of finding the tangentline to some graph of some function atWatch at: 04:20 / 04:40some point which is say X 0 Y Z sothat's the problem that we're addressinghere I guess I should probably turn thisWatch at: 04:40 / 05:00off all right so here's our problem andnow let me show you the solution so welllet's graph the function so let's sayhere's its graph and here's some pointWatch at: 05:00 / 05:20all right maybe I should draw it just abit lower so that I don't all right sohere's a point P maybe it's above thepoint x0 x0 by the way this was supposedto be an x0 that was the some fixedWatch at: 05:20 / 05:40place on the x-axis and now in order toperform this this mighty fee I will useanother color of chalk how about redokay so so here it isWatch at: 05:40 / 06:00there's the tangent line well not quitestraight close enough right I did italright that's the end that's thegeometric problem I achieved what Iwanted to do and it's kind of aninteresting question which unfortunatelyI can't solve for you in this classWatch at: 06:00 / 06:20just how did I do that that is howphysically did I manage to know what todo to draw this tangent line but that'swhat geometric problems are like wevisualize it we can figure it outsomewhere in our brains it happens andthe task that we have now is to figureout how to do it analytically to do itWatch at: 06:20 / 06:40in a way that a machine could do just aswell as I did in drawing this tangentline so so what do we learn in highschool about what a tangent line is wellWatch at: 06:40 / 07:00a tangent line has an equation and anyline through point has the equation yminus y 0 is equal to and the slopetimes X minus X so so here's the theequation for that line and now there areWatch at: 07:00 / 07:20two pieces of information that we'regoing to need to work out what the lineis the first one is the point that'sthat point P there and to specify Pgiven given X we need to know the thelevel of Y which is of course just f ofWatch at: 07:20 / 07:40X 0 now that's that's not a calculusproblem but anyway that's a veryimportant part of the process so that'sthe first thing we need to know and thesecond thing we need to know is theslope and that's this number M and inWatch at: 07:40 / 08:00calculus we have another name for it wecall it f prime of x0 namely thederivative of F so that's the calculuspart that's the tricky part and that'sthe part that we have to discuss now sojust to make that explicit here I'mWatch at: 08:00 / 08:20going to make a definition which is thatf prime of x0 which is known as thederivativeof F at x0 all right is the slope of theWatch at: 08:20 / 08:40tangent line to y equals f of X at thepoint let's just call it P all right soWatch at: 08:40 / 09:00so that's what it is but still I haven'tmade any progress in figuring out anybetter how I drew that line so I have toWatch at: 09:00 / 09:20say something that's more concretebecause I want to be able to cook upwhat these numbers are I have to figureout what this number M is and one way ofthinking about that let me just try itis so I certainly am taking for grantedthat in sort of non calculus part that IWatch at: 09:20 / 09:40know what a line through a point is so Iknow this equation but anotherpossibility might be you know this linehere how do I know unfortunately Ididn't draw it quite straight but thereit is how do I know that this orangeline is not a tangent line but thisother line is a tangent line wellWatch at: 09:40 / 10:00it's it's actually not so obvious andbut I'm going to describe it a littlebit it's it's not really the fact thisthing crosses at some other place whichWatch at: 10:00 / 10:20is this point q but it's not really thefact that the thing crosses it to placebecause the line could be Wiggly thecurve could be Wiggly and it could crossback and forth a number of times that'snot what distinguishes the tangent lineso I'm going to have to somehow graspthis and I first do it in language andWatch at: 10:20 / 10:40it's the following idea it's that if youtake this orange line which is called asecant line and you think of the Q thepoint Q is getting closer and closer toP then the slope of that line will getWatch at: 10:40 / 11:00closer and closer to the slope of thered line and if we draw it close enoughthen that's going to be the correct lineso that's really what I did sort of inmy brain when I drew that first line andso that's the way I'm going toarticulate it first now so the tangentWatch at: 11:00 / 11:20line is equal to the limit of what'sso-called secant lines PQ as Q tends toWatch at: 11:20 / 11:40P and here we're thinking of P is beingfixed and Q is very all right so sothat's the again this is still ageometric discussion but now we're goingWatch at: 11:40 / 12:00to be able to put symbols and formulasto this computation and we'll be able toto work out formulas in any exampleso so let's do that so first of all I'mWatch at: 12:00 / 12:20going to write out these points P and Qagain so maybe we'll put P here and Qhere and I'm thinking of this linethrough them I guess it was orange sowe'll leave it as orange all right andnow I want to compute its slope and soWatch at: 12:20 / 12:40this is gradually we'll do this in twosteps and these steps will introduce usto the basic notations which are usedthroughout calculus includingmultivariable calculus across the boardso the first notation that's used is youimagine here's the x-axis underneath andWatch at: 12:40 / 13:00here's the x0 the location directlybelow the point P and we're travelinghere a horizontal distance which isdenoted by Delta X so that's Delta X socalled and we could also call it theWatch at: 13:00 / 13:20change in X all right so that's onething we want to measure in order to getthe slope of this line PQ and the otherthing is this height so that's thisdistance here which we denote Delta Fwhich is the change in F and then theWatch at: 13:20 / 13:40slope is just the ratio Delta F overDelta X so this is the slope of the ofthe secantand the process I just described overWatch at: 13:40 / 14:00here with this limit applies not just tothe whole line itself but also inparticular to its slope and the way wewrite that is the limit as Delta X goesto zero and that's going to be our slopeso this is the slope of the tangent lineWatch at: 14:00 / 14:20okay now this is still a little a littlegeneral and I'm going to I want to workWatch at: 14:20 / 14:40out a more usable form here I want towork out a better formula for this andin order to do that I'm going to writeDelta F the numerator more explicitlyhere the change in F so remember thatthe point P is the point X 0 f of X 0Watch at: 14:40 / 15:00alright that's what we got from ourformula for the point and in order tocompute these distances in particularthe vertical distance here I'm going tohave to get a formula for Q as well soWatch at: 15:00 / 15:20if this horizontal distance is Delta Xthen this location is X 0 plus Delta Xand so the point above that point has aformula which is X 0 plus sorry plusDelta X F of and this is a mouthful X 0Watch at: 15:20 / 15:40plus Delta X all right so there's theformula for the point Q here's theformula for the point P and now I canwrite a different formula for theWatch at: 15:40 / 16:00derivative which is the following sothis F prime of X 0 which is the same asM is going to be the limit as Delta Xgoes to 0 of the change in F well theWatch at: 16:00 / 16:19change in F is the value of F at theupper point here which is x0 plus DeltaX and minus its value at the lower pointP which is f of x0 divided by Delta XWatch at: 16:19 / 16:40all right so this is the formula I'mgoing to put this in a little boxbecause this is by far the mostimportant formula today which we use toderive pretty much everything else andthis is the way that we're going to beable to compute these numbers so let'sWatch at: 16:40 / 17:00let's do an exampleWatch at: 17:00 / 17:20this example so we'll call this exampleone we'll take the function f of X whichis 1 over X that's sufficientlyWatch at: 17:20 / 17:40complicated to have an interestinganswer and sufficiently straightforwardthat we can compute the derivativefairly quickly so so what is it thatwe're going to do here all we're goingto do is we're going to plug in thisWatch at: 17:40 / 18:00this formula here for that functionthat's that's all we're going to do andvisually what we're accomplishing issomehow to take the hyperbola and take apoint on the hyperbola and figure outsome tangent line all right that's whatWatch at: 18:00 / 18:20we're accomplishing when we do that sowe're accomplishing this geometricallybut we'll be doing it algebraically sofirst we consider this difference DeltaF over Delta X and write out its formulaso I have to have a place so I'm goingto make it again above this point X 0Watch at: 18:20 / 18:40which is a general point we'll make thegeneral calculation so the value of F atthe top when we move to the right by fof X so I just read off from this readoff from here the formula the firstthing I get here is 1 over X 0 plusWatch at: 18:40 / 19:00Delta X that's the left-hand term minus1 over X 0 that's the right-hand termand then I have to divide that by DeltaX ok so here's our expression and by theway this has a name this thing is calledWatch at: 19:00 / 19:20a difference quotientit's pretty complicated because there'salways a difference in the numerator andin disguise the denominator is adifference because it's the differencebetween the value on the right side andWatch at: 19:20 / 19:40the value on the left side here okay sonow we're going to simplify it by somealgebra so let's just take a look sothis is equal to let's continue on thenext level here this is equal to 1 overWatch at: 19:40 / 20:00Delta x times now all I'm going to do isput it over a common denominator so thecommon denominator is X 0 plus Delta Xtimes X 0 and so in the numerator forthe first expression I have X 0 and forWatch at: 20:00 / 20:20the second expression I have X 0 plusDelta X so this is a the same thing as Ihad in the numerator before factoringout this denominator and here I put thatnumerator into this more amenable formand now there are two basiccancellations the first one is that X 0Watch at: 20:20 / 20:40and X 0 cancel so we have this and thenthe second step is that these twoexpressions cancel right the numeratorWatch at: 20:40 / 21:00and denominator now we have acancellation that we can make use of sowe'll write that under here and this isequals minus 1 over X 0 plus Delta Xtimes X 0 and then the very last step isWatch at: 21:00 / 21:20to take the limit as Delta X tends to 0and now we can do it before we couldn'tdo it why because the numerator and theDominator gave us 0 over 0 but now thatI've made this cancellation I can passto the limit and all that happens is IWatch at: 21:20 / 21:40set this Delta X equal to 0and I get minus 1 over X zero squaredalright so that's the answerall right so in other words what I'veshown when we put it up here is that fprime of X zero is minus 1 over X 0Watch at: 21:40 / 22:00squared now let's let's look at thegraph just a little bit to check thisfor plausibility all right what'sWatch at: 22:00 / 22:20happening here is first of all it'snegative right it's less than 0 which isa good thing you see that slope there isnegative that's the simplest check thatyou could make and the second thing thatWatch at: 22:20 / 22:40I would just like to point out is thatas X goes to infinity that as if as wego farther to the right it gets less andless steep so less and whoops as X go x0goes to infinity not not 0 as X 0 goesWatch at: 22:40 / 23:00to infinity less and less steep sothat's also consistent here if when X 0is very large this is a smaller andsmaller number in in magnitude althoughit's always negative it's always slopingdown all right so I've managed to fillWatch at: 23:00 / 23:20the boards so maybe I should stop for aquestion or two yesso the question is to explain again thisWatch at: 23:20 / 23:40limiting process so the formula here iswe have basically two numbers so inother words why is it that thisexpression when Delta X tends to 0 isequal to minus 1 over X zero squared letme let me illustrate it by sticking in anumber for X 0 to make it more explicitalright so for instance let me stick inWatch at: 23:40 / 24:00here for X 0 the number 3 then it'sminus 1 over 3 plus Delta X times 3that's the situation that we've got andnow the question is what happens is thisnumber gets smaller and smaller andsmaller and gets to be practically 0Watch at: 24:00 / 24:20well literally what we can do is justplug in 0 there then you get 3 plus 0times 3 in the denominator minus 1 inthe numerator so this tends to tends tominus 1 over 9 over 3 squared and that'swhat I'm saying in general with thiswith this extra number here other otherWatch at: 24:20 / 24:40questions yesso the question is how what happenedWatch at: 24:40 / 25:00between this step and this step rightexplain this this step here alright sothere were two parts to that the firstis this Delta X which is sitting in thedenominator I factored all the way outfront and so what's in the parenthesesis supposed to be the same as what's inWatch at: 25:00 / 25:20the numerator of this other expressionand then at the same time as doing thatI put that expression which is adifference of two fractions I expressedit with a common denominator so in thedenominator here you see the product ofthe denominators of the two fractionsand then I just figured out what thenumerator had to be without really yeahWatch at: 25:20 / 25:40other questions okay so now so I claimthat on the whole calculus is gets a badWatch at: 25:40 / 26:00rap that it's actually easier than mostthings but it has there's a perceptionthat it's that it's that it's harder andso I really have a duty to to give youthe calculus made harder story here sowe have to make things harder becauseWatch at: 26:00 / 26:20that's that's our job and this is whatactually what most people do in calculusand it's the reason why calculus has abad reputation so the secret is thatwhen people ask problems in calculusthey generally ask them in context andthere are many many other things goingWatch at: 26:20 / 26:40on and so the little piece of theproblem which is calculus is actuallyfairly routine and has to be isolatedand gotten through but all the rest ofit relies on everything else you learnedin mathematics up to this stage fromgrade school to through high school soso that's the complication so now we'reWatch at: 26:40 / 27:00going to do a little bit of calculusmade hardby talking about a word problem now wewe only have one sort of word problemthat we can pose because all we'vetalked about is this geometry point ofWatch at: 27:00 / 27:20view so so far those are the only kindsof word problems we can pose so whatwe're going to do is just pose such aproblem so find the areas of trianglesenclosed by the axes and the tangent toWatch at: 27:20 / 27:40y equals 1 over okay so that's aWatch at: 27:40 / 28:00geometry problem and let me draw apicture of it it's practically the sameas the picture for example one of courseso here's we're only consider the firstquadrant here's our shape all right it'sthe hyperbola and here's maybe one ofWatch at: 28:00 / 28:20our tangent lines which is coming inlike this and then we're trying to findthis area here all right so there's ourproblem so why does it have to do withcalculus it has to do with calculusbecause there's a tangent line in it andso we're going to need to do someWatch at: 28:20 / 28:40calculus to to answer this question butas you'll see the calculus is the easypart so so let's get started with thisproblem first of all I'm going to labela few things and one important thing toremember of course is that the curve isWatch at: 28:40 / 29:00y equals 1 over X that's perfectlyreasonable to do and also we're going tocalculate the areas of the triangles andyou could ask yourself in terms of whatwell we're going to have to pick a pointand give it a name and since we need anumber we're going to have to do morethan geometry we're going to have to doof this analysis just as we've doneWatch at: 29:00 / 29:20before so I'm going to pick a point andconsistent with the labeling we've donebefore I'm going to call it x0 y0 sothat's almost half the battlehaving notations x and y for thevariables and x0 and y0 for the for thespecific point now once you see that youWatch at: 29:20 / 29:40have these labelingsI hope it's reasonable to do thefollowing so first of all this is thepoint x0 and over here is the point y0that's something that we're used to ingraphs and in order to figure out thearea of this triangle it's pretty clearWatch at: 29:40 / 30:00that we should find the base which isthat we should find this location hereand we should find the height so we needto find that value there all right solet's let's go ahead and and do it sohow are we going to how are we going toWatch at: 30:00 / 30:20do this wellso let's let's just take a look so whatis it that we need to do I claim thatthere's only one calculus step and I'mWatch at: 30:20 / 30:40going to put a star here for thistangent line I have to understand whatthe tangent line is now once I figuredout what the tangent line is the rest ofthe problem is no longer calculus it'sjust that slope that we need so what'sthe formula for the tangent line putthat over here it's going to be Y minusWatch at: 30:40 / 31:00y 0 is equal to and here's the magicnumber we already calculated it it's inthe box over there it's minus 1 over X 0squared X minus X 0 so this is the onlybit of calculus in this problem but nowWatch at: 31:00 / 31:20we're not done we have to finish it wehave to figure out all the rest of thesequantities so we can figure out the areaWatch at: 31:20 / 31:40all right so how do we do that well toWatch at: 31:40 / 32:00find this point this has a name we'regoing to find the so-called x-interceptthat's the first thing we're going to doso to do that what we need to do is tofind where this horizontal line meetsWatch at: 32:00 / 32:20that diagonal line and the equation forthe x-intercept is y equals 0 all rightso we plug in y equals 0 that's thishorizontal line and we find this pointso let's do that in 2 star so we get 0minus oh one other thing we need to knowWatch at: 32:20 / 32:39we know that y 0 is f of x 0 and f of Xis 1 over X so this thing is 1 over X 0right and that's equal to minus 1 over X0 squared and here's X and here's X 0Watch at: 32:39 / 33:00all right so in order to find this xvalue I have to plug in one equationinto the other so this simplifies a bitlet's put let's see this is minus x overWatch at: 33:00 / 33:20x0 squared and this is plus 1 over X 0because the x0 and x0 squared cancelssomewhat and so if I put this on theother side I get X divided by x0 squaredis equal to 2 over X 0 and if I thenWatch at: 33:20 / 33:39multiply through so that's what thisimplies and if I multiply through by x0squared I get X is equal to 2 X 0 okokay so I claim at this point we've justWatch at: 33:39 / 34:00calculated it's 2 X 0now I'm almost done I need to get theWatch at: 34:00 / 34:20other one I need to get this one up herenow I'm going to use a very big shortcutto do that so so the shortcut to they-intercept sorry yeah the y-interceptis to use symmetry alright I claim I canWatch at: 34:20 / 34:40stare at this and I can look at that andI know the formula for the y-interceptit's equal to 2 y 0 all right that'sWatch at: 34:40 / 35:00what that one isso this one is 2 y 0 and the reason Iknow this is the following so here's thesymmetry of the situation which is notcompletely direct it's a kind of mirrorsymmetry around the diagonal it involvesthe exchange of X Y with y X so tradingWatch at: 35:00 / 35:20the roles of X and y so the symmetrythat I'm using is that any formula toget that involves X's and Y's if I tradeall the x's and replace them by Y's andtrade all the Y's and replace them byX's then I'll have a correct formula onthe other way so everywhere I see a whyWatch at: 35:20 / 35:40I'm making an X and everywhere I see anX I make it a y this which will takeplace so why is that that's because thethat's just an accident of this equationthat's becauseso the symmetry explained is that theWatch at: 35:40 / 36:00equation is y equals 1 over X but that'sthe same thing as X y equals 1 if Imultiply through by X which is the samething as x equals 1 over y so here'swhere the X and the y get reversed okayWatch at: 36:00 / 36:20now if you don't trust this explanationyou can also get get the y intercept byplugging x equals 0 into the into theWatch at: 36:20 / 36:40equation star okay we plugged y equals 0in and we got the x value and you coulddo the same thing analogously the otherway alright so I'm almost done with theWatch at: 36:40 / 37:00with the geometry problem and let'slet's finish it off now well let me holdoff for one second before I finish itWatch at: 37:00 / 37:20off what I'd like to say is just makeone more tiny remark right and this isthe hardest part of calculus in myopinion so the hardest part of calculusis that we call it one variable calculusbut we're perfectly happy to deal withWatch at: 37:20 / 37:40four variables at a time or five or anynumber in this problem I had an X a Yand X 0 and a wiser that's already fourdifferent things they have variousinterrelationships between them so ofcourse the manipulations we do with themare algebraic and when we're doing thederivatives we just consider one what'sWatch at: 37:40 / 38:00known as one variable calculus butreally there are millions of variablesfloating around potentially so that'swhat makes things complicatedcreated and that's something that youhave to get used to now there'ssomething else which is more subtle inthat I think many people who teach thesubject or use the subject aren't awarebecause they've already entered into theWatch at: 38:00 / 38:20language and they're not they're socomfortable with it that they don't evennotice this confusion there's somethingdeliberately sloppy about the way wedeal with these variables the reason isvery simple there are already fourvariables here I don't want to createsix names four variables or eight namesWatch at: 38:20 / 38:40four variables and but really in thisproblem there were about eight I justslipped them by you so why is that wellnotice that the first time that I got aformula for Y zero here it was thispoint and so the formula for y zeroWatch at: 38:40 / 39:00which I plugged in right here was fromthese the equation of the curve y 0equals 1 over X 0 the second time I didit I did not use y equals 1 over X Iused this equation here so this is not yWatch at: 39:00 / 39:20equals 1 over X that's the wrong thingto do that's an easy mistake to make ifthe formulas are all a blur to you andyou're not paying attention to wherethey are on the diagram you see thaty-intercept of that x-interceptcalculation they're involved where thishorizontal line met this diagonal lineWatch at: 39:20 / 39:40and y equals 0 represented this linehere so the sloppiness is that Y meanstwo different things and we do thisconstantly because it's way way morecomplicated not to do what concept to doit it's much more convenient for us toallow ourselves the flexibility toWatch at: 39:40 / 40:00change the role that this letter playsin the middle of the of the computationand similarly later on if I had donethis by this more straightforward methodfor the y-intercept I would have set Xequal to zero that would have been thisvertical line which is x equals zero butWatch at: 40:00 / 40:20I didn't change the letter X when Ithat because that would be a waste forus so this this is this is one of themain confusions that happens if you cankeep yourself straight you're you're alot better off and and as I say this isthis is this is one of the complexitiesWatch at: 40:20 / 40:40all right so now let's finish off theproblem on let me finally get this areahere so actually I'll just finish it offright here so the area of the triangleis well it's the base times the heightWatch at: 40:40 / 41:00the base is 2 X 0 the height is 2 y 0and a half of that so it's 1/2 2 X 0 x 2y 0 which is 2 X 0 y 0 which is low andbehold - so the amusing thing in thiscase is it actually didn't matter what XWatch at: 41:00 / 41:200 and y 0 are we get the same answerevery time that's just an accident ofthe function 1 over X happens to be thefunction with that property alright soWatch at: 41:20 / 41:40we have still have more business todayserious business so let me continue sofirst of all I want to give you a fewmore notations and these are just otherWatch at: 41:40 / 42:00ways that people refer notations thatpeople use to refer to derivatives andthe first one is the following wealready wrote y is equal to f of X andso when we write Delta Y that means theWatch at: 42:00 / 42:20same thing is Delta F that's a typicalnotation and previously we wrote F primefor the derivative so this is this isthis is Newton's notation for thederivative okay but there are otherWatch at: 42:20 / 42:40notations and one of them is DF DX andanother one is dy DX meaning exactly thesame thing and sometimes we let thefunction slip down below so that becomesd by DX of F or D by DX of Y so theseWatch at: 42:40 / 43:00are all notations that are used for thederivative and these were initiated bylive notes and these notations are usedinterchangeably sometimes practicallytogether they both turn out to beextremely usefulthis one omits notice that this thingWatch at: 43:00 / 43:20omitsthe underlying base point x0 that's oneof the nuisances it doesn't give you allthe information but there are lots ofsituations like that we're where peopleleave out some of the importantinformation you have to fill it in fromWatch at: 43:20 / 43:40context so that's another couple ofnotations so now I have one morecalculation for you today I carried outthis calculation of the derivative ofthe of the the derivative of theWatch at: 43:40 / 44:00function 1 over X I want to take care ofsome other powers so let's do thatso example 2 is going to be the functionWatch at: 44:00 / 44:20f of X is X to the N N equals one twothree one of these guys and now whatwe're trying to figure out is thederivative with respect to X of X to theWatch at: 44:20 / 44:40N in our new notation what this is equalto so again we're going to form thisexpression Delta F Delta X and we'regoing to make some algebraicsimplification so what we plug in forWatch at: 44:40 / 45:00Delta F is X plus Delta X to the N minusX to the N divided by Delta X now beforelet me just stick this in and I'm goingto erase it before I wrote x0 hereand x0 there but now I'm going to getrid of it because in this particularWatch at: 45:00 / 45:20calculation it's a nuisance I don't havean X floating around which meanssomething different from the x0 and Ijust don't want to have to keep onwriting all those symbols it's a wasteof blackboard energy there's a totalamount of energy that I'm you know I'vealready filled up so many blackboardsthat it's just a limited amount of plusWatch at: 45:20 / 45:40I'm trying to conserve chalk okay anywayno zeros so think of X is fixed again inthis case Delta X moves and X is fixedin this in this calculation all rightWatch at: 45:40 / 46:00now in order to simplify this in orderto understand it algebraically what'sgoing on I need to understand what thenth power of a sum is and that's afamous formula we only need a littletiny bit of it called the binomialtheorem so the binomial theorem by noWatch at: 46:00 / 46:20mealtheorem which is in your text andexplained in a in an exercise says in anappendix sorry says that if you take thesum of two guys and you take them to thenth power that of course is X plus DeltaX multiplied by itself and times and soWatch at: 46:20 / 46:40the first term is X to the N that's whenall of the N factors come in and thenyou could have this factor of Delta Xand all the rest X's so at least oneterm of the form X to the N minus oneWatch at: 46:40 / 47:00times Delta X and how many times doesthat help well it happens when there's afactor from here from the next factorand so on and so on and so on there's atotal of n possible times that thathappens and now the great thing is thatwith this alone all the rest of theWatch at: 47:00 / 47:20terms are junk that we won't have toworry aboutso to be more specific the junk there'sa very careful notation for the junk thejunk is what's called Big O of Delta xsquared what that means is that theseWatch at: 47:20 / 47:40are terms of order so with Delta xsquared Delta X cubed or higheralright that's how very excitingWatch at: 47:40 / 48:00higher-order terms okay so this is theonly algebra that we need to do and nowwe just need to combine it together toget our result so now I'm going to justcarry out the cancellations that we needWatch at: 48:00 / 48:20so here we go we have Delta F over DeltaX which remember was 1 over Delta Xtimes this which is this x now this is XWatch at: 48:20 / 48:40to the n plus N X to the N minus 1 DeltaX plus this junk term minus X to the Nall right so that's what we have so farbased on our previous calculations nowWatch at: 48:40 / 49:00I'm going to do the main calccancellation which is this alright sothat's 1 over Delta X times n X to the nminus 1 Delta X plus this term here andWatch at: 49:00 / 49:20now I can divide in by Delta X so I getn X to the N minus 1 plus now it's o ofDelta X there's at least one factor ofDelta X naught 2 factors of Delta Xbecause I have to cancel one of them andnow I can just take the limit in theWatch at: 49:20 / 49:40limit this term is going to be 0 that'swhy I called it junk originally becauseit disappears and in math junk issomething that goes away so this tendsto as Delta X goes to 0 and X to the Nminus 1 and so what I've shown you isthat D by DX of X to the N minus sorry nWatch at: 49:40 / 50:00is equal to n X to the N minus 1 so nowthis is going to be super important toyou right on your problem set in everypossible way and I want to tell you onething one way in which it's veryimportant in one way that extends itWatch at: 50:00 / 50:20immediately so this thing extends topolynomialswe got quite a lot out of this onecalculation namely if I take D by DX ofsomething like X cubed plus 5x to theWatch at: 50:20 / 50:4010th power that's going to be equal to3x squared that's applying this rule toX cubed and then here I'll get 5 times10 so 50 x to the 9th so this is thetype of thing that we get out of itand we're going to make more hay withthat next time what I turn myself offWatch at: 50:40 / 51:00yes the question is the question was theWatch at: 51:00 / 51:20binomial theorem only works when X DeltaX goes to 0 no the binomial theorem is ageneral formula which also specifiesexactly what the junk is it's very muchmore detailed but we only needed thispart we didn't care what all these crazyterms were it's it's it's junk for ourWatch at: 51:20 / 51:40purposes now because we don't happen toneed any more than those first two termsyes because the Delta X goes to 0 ok seeyou next time