Lecture 05: Definition of the Derivative

February 16, 2015

Start | about the limits and the tangent problem and how to find the slope of the curve so professor sutherland put up a little blurb on this but this is the definition of the derivative in other words how we actually find the slope of an actually vurve so what do we mean when we say the slope of the curve, the slope of the curve is curves do this right |

0:30 | so i guess i have to warm that chalk up a bit so you have to figure out where on the curve we are talking about because at any point the slope of the line is different so we have a cuve it looks like that, the slope is sort of flat here pretty steep here the one thing we did is what did we mean by the slope of the curve you could imagine that you have a line |

1:01 | tangent to the curve at the sport where the line is tangent to the curve thats what we mean by the slope of the curve, we mean the slope of the line so this line has a prettyy flat slope pretty close to 0 so we would say the slope of the curve here is pretty close to 0 there thats a slope of about 1 heres where it gets pretty steep do lets say thats 5, i just made that up i actually have no idea what it is |

1:34 | so the slope of the curve is about 5
so how are we going to calculate these things?
well lets say we have lets say we have f of x equals, lets make one thats very simliar to webassign 7x- x squared you want to find the slope |

2:01 | at the point 2,10 so what does 7x-x squared look like well you can factor this that says x times 7-x if you cant read what i wrote there, thats not important but you want to know how to graph it it has a 0 at the origin |

2:32 | a 0 at 7\ upside down parabula so it kind of looks like that and we want to find the slope at point 2 comma 10 and we want to know and we want to know what the slope of that, its a positive answer so if you get a negative you know its wrong so im gonna redraw that graph in a second, im gonna blow it up |

3:01 | its easier to see so we want the slope right about there 2,10 so the problem is when you find slope its differences in y's and differences in x's |

3:35 | okay so slope is y2-y1/x2-x1 but we only have 1 point, 2/10 so how would we find the slope at exactly 2,10 what we can do is pick another point so remember we did this 2.1,2.01 and so on so lets just call this other point |

4:03 | 2 plus h comma something can you tell thats a 2? is really hard to write with this chalk so then the slope would be the y value at 2 minue the y value at 2 plus h over 2+h or the other way around |

4:31 | so were gonna want to find f of 2 plus h minus f of 2 over 2 plus h minus 2, that would give us close to the slope that would make h a really small number thats what we were doing last week, you can make h point 1 you can make it .001 what were gonna do is were gonna use limits and make h 0, the limit 0, remember |

5:00 | its not actually 0 its infinitly close to 0 but at some point we dont can and we say well its 0 so were gonna do this and were gonna take the limit as h goes to 0 so how do we find f of 2+h, you plug 2+h in the equation |

5:33 | so that is 7 times 2 plus h minu 2 plus h squared that is f of 2+h if it was f of 5 you plug in 5 f of 10 you plug in 10 f of 2 you plug in 2+h you plug in 2 plus h and your plugging i in to here okay so where you se x your plugging it in with 2+h |

6:03 | so thats f of 2+h we already know f of 2 f of 2 is 14 minus 4 is 10 and 2 plus h minus 2 is h greater now lets just plug in 0 you plug in 0 the problem is you get 0/0 so were now gonna have to rearrange this somehow see with some magic of algebra we can get an answer |

6:32 | im gonna have some room issues but lets see what i can do this is 7 times 2 is 14 plus 7h minus, okay can you guys multiply out 2 plus h lets do it over here |

7:05 | thats easy with the foiling process
so that is
4+4h
plus h squared minus 10
all over h
have you been watching the videos by the way?
many of you are saying yes but we know how many of you are actually watching the video |

7:30 | i recommend you do especially as you get close to the exam you may to go through everything a second time or not you may have better things in life, once is good enough i recommend you watch it so we have 14 plus, now lets simplify that limit as h goes to 0 14+7h |

8:00 | minus 4 minus 4h minus h squared be careful when you distribute that minus sign it may go negative when its suppose to be negative typical place to mess up be very careful with your minus signs minus 10\ over h the problem is at every satge we plug in h equals 0 were gonna keep getting 0/0 but now we can do some simplify 14 minus 4 is 10 minus 10 is 0 |

8:41 | we get 7h minus 4h minus h2 wait 7h minus 4h is 3h now factor out the h |

9:04 | and we can cancel those h's how come we dont cancel out the h cause we have h/h when you plug in h equals 0 you cant cancel 0/0 remmeber they taught you that in about 7th or 8th grade we can cancel those because h is not actually 0 but the limit of h approaches 0 h is an infinitely small number but it aint 0 so the we dont have to cancel it and now we say that if h gets close to 0, this gets close to 3 |

9:35 | so if you want to know the slope of the parabula is at x equals 2 , its 3 and of course you say h isnt actually 0 so its not actually 3 so you say yes but its close to 3 as you want it to be so you might as well say its 3 that ius how you do the derivative everyone understand what i did? its just slope, im looking at the tangent line |

10:04 | i want to find the slope of this line so i use difference in ys f 2+h and f of 2 over difference of x's 2+h and 2 and then the limit i shrink h to be 0 how do i do that? i multiple 2+h in the equation i multiply everything out and do lots of canceling and when i get all done i get 3 |

10:32 | easy for me to do, lets do another one |

11:15 | okay were gonna find the derivative of f equals 3x squared plus 1 as x equals 2 so how do we do this? were gonna do just like we did before |

11:34 | were gonna have a curve thats a parabula what does 3x squared plus 1 look like its just a parabula and goes down to 1 how do i know it goes down to 1, i plug in 0 and i get 0 plus 1 and im doing it at 2 a little something like that, at x equals 2 |

12:08 | just what i did before i use 2 and we use the point 2 plus h and im gonna squeeze them together untill i get the derivative so im gonna do it doesnt always have to be 2 by the way |

12:30 | f of 2+h minus f of 2 all over h as h goes to 0, thats what im looking for so first i need to figure out what is f of 2+h and what is f of 2 |

13:01 | f of 2+h is what i get when i plug in 2+h for x so it would equal 3 times 2+h squared plus 1 so this is gonna be plug in 2+h what is f of 2 f of 2 is what you get when you plug in just 2 |

13:34 | whcih is 13 by the way how do i know its h on the bottom i sort of jumped at that step i know its h because its the difference between 2+h and 2 the bottom is always going to be h |

14:01 | alright lets figure out what f of 2+h is, lets multiply that out 2+h squared is 4+4h+h squared plus 1 so you get 12 +12h plus h squared plus 1 well this becomes |

14:34 | 12 plus 12h plus h squared plus 1 minus 13 all over h oh its 3h squared youre right sorry |

15:01 | course thats not gonna matter when we get to the end okay if you cant read that well fix that thats much better so what do i do? 12 plus 1 is 13 minus 13 is 0 1h plus 3h squared |

15:31 | all over h okay factor h out of the top h over 12 plus 3h all over h the h's cancel through the magic of alegbra and now when i plug on h is 0 i get 12 |

16:04 | so if you want to know the slope of this line at x equals 2 you get 12 lots of puzzled faces |

16:32 | the final answer would be 12 nope the question is do we have to show the limit of anything nope the answer of the quesitojn what is the derivative of plus 3x squared plus 1 and x equals 2 the answer is 12 exceot its not its 13 right no its 12 im sorry its 12 |

17:02 | confusing myself its 12 ah the questions is why do we do it when h approaches 0, so remember were trying to fin the slope of this line and the slope of this line we find by doing difference in y over difference of x so i pick 2, 2 and 2plus h i get the coresponding y values for the slope the coresponding y values are f of 2 and f of 2+H |

17:32 | and then i want to get the slope of a line a certain point i really dont want 2 ppoints, i only want 1 point so i want h to get as close to 0 as possible well 2 and 2+h are the same thing thats really what im trying to do, trying to get on top of each other so as h gets closer and closer to 0 these 2 x values get closer and closer together until its just 2 and thats the miracle of limis |

18:06 | should i do another one? yea okay so im gonna write it down and you guys are gonna try it on your own first and then well do it as a team so ill put the problem up here |

18:54 | what we want to do is find the limit |

19:00 | when h gets really small so to 0 but the function at 5, so 5 plus h minus the function by its self and the bottom is h minus 5 so the difference in y's there difference in the x's, were gonna have 2 x's value were gonna have a value at 5 an x value at x+h, and a y value at 5 and a y value at 5+h |

19:30 | and now you just have to do some algebra so the limit h goes to 0 plug in 5+h, you get 5+H squared plus 3 times 5 plus h remember plug in 5 plus h for both terms typical stuff they mess this up on the exam they only plug in for the first x |

20:00 | plug in for all of the x's once each minus f of 5, f of 5 is 25 plus 15 is 40 or yu can write the 25 and 15 but then dont forget to distribute the minus sign cause minus 25 minus 15 5 plus h minus 5 is h, you should get h on the bottom if you didnt get h on the bottom you messed up another tip when you do these derivatives |

20:30 | and this is a polynomial like x squared plus 3x and you work this out, all the terms on the right cancel all the terms on the right that didnt happen you did something incorrectly on the right i hae 40 so somewhere this is gonna have to give you 40 and there gonna have to cancel alright lets do this 5 plus h squared is 25 |

21:00 | plus 10h plus h squared plus 15 plus 5h minus 40 all over h did i make a mistake so far 3h |

21:37 | see when we shoot the movie version of this will edit that out, well replace it well get a stunt devil and you can vote who you want to portrey me 25 plus 15 minus 40 they cancel |

22:05 | and you get 10h plus h squared plus 3h you can make that 13h right away so this is as h goes to 0, 13h plus h squared all over h factor out an h, see how we do it the exact same way everytime you get this you take out an h you cancel this is gonna happen with polynomials |

22:40 | cancel the h's now when you get the limit you get 13 so the answer we are looking for is 13 were not looking for x equals 13 or f of something is 13 were just looking for 13 how can we make this a little harder cause this isnt hard enough lets make it harder |

23:03 | what if instead of doing it at x equals 5 i know when i redo this problem i want to do x=a seriously a i was just getting the hang of this stuff what if we do a greek letter like alpha |

23:31 | i hope you guys wrote that down cause its gone, not lets do it with a so now im gonna do, f of a plus h okay we got it a plus h minus f of a over a+h minus a |

24:00 | its the same technique by now theres no numbers to make your life easy so first lets plug in the variance things and lets see what happens what is f of a+h, have to take x and replace it with a+h so i get a+h squared plus 3 a+h minus when i plug in a i get a squared |

24:32 | plus 3a on the bottom i get h remember the tip i just gave you these terms on the right are gonna have to cancel the terms on the left so after your done with canceling every single thing that contain an h term on top then youll get to pull that h out and cancel the h on the bottom if you do that you can get it correctly |

25:00 | so youll know you messed up the problem if these terms dont end up canceling these terms usually means you messed up a minus sign thats why you should do your exam in pencil when you do it in pen and you mess up your paper would be a big indecisive the tas would not care for what you do maybe they will but i wouldnt take that chance |

25:32 | alright lets work that out you get limit as h goes to 0 a plus h squared is a squared plus 2ah plus h squared plus 3a plus 3h minus a squared minus 3a |

26:02 | all over h okay you can cancel a squared cancel and the 3a cancels just the way i promise im left with 2h plus h squared plus 3h |

26:31 | all over h and now look i can pull an h out of everything i pull an h out of everything and i get 2a plus h plus 3 all over h cancel the h's |

27:04 | and now when h goes to 0 i get 2a plus 3 so that say that the derivative of f of x equals x squared plus 3x at a point a this h i do the limit, so what happens to this h when h goes to 0 |

27:37 | so this says at any particular point at a the derivative should be 2a+3 lets check it out, when we have 5 what do we get we have 2 times 5 is 10 pllus 3 is 13 so now were convinced doesnt mean your right but your more convinced so far so good? and how can we make this nastier |

28:01 | oh we dont have to give you a polynomial then you can do something fun ill wait till everyone has this down and ill do a fun one you ready for me to start erasing want to try it on your own first or do it as a team on our own |

28:39 | find the derivative of f of x equals 2/x+5 and at x equals 3 |

29:03 | not as bad as it looks its just to test your algebra i dont know if there is gonna be any other review session it may just be 1 were gonna have class next week monday and wednesday so thats lots of reviewing and that much in clas alright how do we do this?use our formula, find our limit |

29:31 | as h goes to 0 2/ 3 plus h plus 5 well im gonna move this sorry so im gonna take x and replace it with 3+h and replace it with 3 and divide by 3+h 3 so f of 3+h |

30:01 | minus f of 3 if you want to get particle credit on the exam make sure you write that somewhere in case you mess up at least we know what youre doing well sorting know what youre doing try and leave a trail its okay if you take the big step over h i recognized that many of you take calculus so you know where this is goign |

30:31 | thats good so this is 2/3+h+5 minus 2/3+5 all over h which we can simplify to h goes to 0 2/8+h minus 2/8 |

31:01 | all over h so how can we combine those common denominator, and by the way you can reduce the 2/8 to 1/4 it doesnt matter so the limit as h goes to 0 im gonna take the term on the left and multiply top and bottom by h the left and multiply it by 8 the term on the right and multiply it by 8+h |

31:32 | so im gonna have 2/8+h times 8/8 minus 2/8 times 8+h/8=H |

32:01 | all over h how are we at this part? people okay with simplifying so this becomes 16 over 8 plus h times h and this is 16 plus 2h |

32:37 | the whole thing is over h, watch the minus sign very easy to mess up now i have the limit as h goes to 0 16 minus 16 minus 2h |

33:00 | all over 8+h times 8 all over h the 16's cancel you get minus 2h over 8+h |

33:31 | times 8 all over h now now h on the top expression can cancel with this h why cauce this is really h/1 so you canc flip the fraction and multiply so you get the limit h goes to 0 minus 2h |

34:04 | you get that minus 2h over 8+h times 8 times the 1/h so this h down here is gonna cancel with that h now when you plug in 0 your not gonna have a problem anymore youre gonna get -2 on top 8 and 8 is 64 you can reduce that or not -1/32 |

34:32 | got the idea lets do one more make sure everyone gets it this hard enough, maybe i should do one harder this h this h when you do the limit as h goes to 0 okay lets give you one more type |

35:00 | make sure you get it lets find the derivative of the square root of x plus 1 at x equals 8 alright we ready to do this as a team |

35:55 | so this time we want f at 8+h |

36:01 | minus f of 8 over 8+h minus 8 thats gonna be the limit as h goes to 0 the square root of 8+h+1 minus the square root of 8+1 which is 3 all over h |

36:30 | and 8+1 is 9 so this is gonna be the limit as h goes to 0 square root of 9 plus h minus 3 all over h what do we do we multiply the top and bottom by the conjagate remember when you see the square root thinkg conjugate the conjugate will be the square root of 9 plus h |

37:00 | plus 3 why did we do that cause its gonna work |

37:30 | okay so the top becomes the square root of 9 plus h times the square root of 9 plus h is 9+h the point of the conjugate is the two middle terms cancel because you get 3 minus radical 9+h minus 3 radical 9+h so they cancel and then you just get 3 times 3 is 9 |

38:03 | and when yu do this dont do anything with the terms on the bottom the 9's cancel and on top you just have an h and on the bottom you have h times the square root of 9+h +3 |

38:30 | and those h's cancel aand youre left wit 1 not 0 the limit as h goes to 0, you have 1 over the square root of 9+h plus 3 now we can let h go to 0 because we dont have a problem with 0/0 anymore we have 1 over the square root of 9 plus 3 which is 1/6 |