Lecture 07: The power rule & some review

February 23, 2015

Start | you can admit it, you can still come to class the video okay you go to the website see where it says i should send this to my mom she would be so happy |

0:31 | well happy is a small word my father was watching he already give me constructive criticism he taught here for 40 years or so if any of you wonder through the c level of the physics building at the library, it would say kahn, math physic elabarate, not the name for me |

1:00 | alright so any of you have friends that already know calculus?
your sitting next to them right now in class and theyre really annoying to have next to you cause they kind of know all this stuff they look at you and go thats 7 x too, and your like now youre gonna learn exactly whats going on so youre gonna learn the power rule i talk to professor sutherland for those people who raise their hands and say can we use the short cut and the answer is yes depending on the question i know that makes a lot of you feel really happy |

1:31 | suppose you want to do were gonna find the derivative of that using the definition im not gonna do this the long way but thats the definition of the derivative some of you are already bored |

2:02 | now what happens when you multiply out x plus h, you get you get x squared plus 2xh plus h squared minus x squared over h do you remember what i told you about a polynomial these terms on the right will always cancel the terms on the left that first term you foil this out the x squared thats gonna cancel with the x squares |

2:34 | all the remaining terms have at least one power in each of them this has an h this has an h squared so now youll be able to pull out an h you get 2x plus h over h now when you take the limit everything that has an h left in it is going to b 0 because when you do h goes to 0 |

3:01 | the h is 0, everything without an h in it stays thats the derivative so far so good thats easy lets do it again with cubed f of x of x cubed now remember that x cubed term is gonna end up canceling |

3:30 | because when you multiply out x plus h cubed that first term, you got x plus h times x plus h imes x plus h you multiply that out and you get a term thats x times x times x which is x cubed that term is gonna cancel with this x cubed the other terms are all gonna have some h term in there once more than once, so well multiply that out |

4:01 | you get x cubed plus 3x squared h plus 3xh squared plus h cubed minus x cubed over h the x cubed cancel as i said everything else has an h in it with more than one power of h, but at least 1 h |

4:30 | so you can pull the h out and youre left with 3x squared that does not have h anymore but all the other terms still have at lest 1 power of h we know where this is going, the h's cance; and now everything that has so term of h in it you let h go to 0, all those terms are gonna go to 0 |

5:02 | only thing left is 3x squared so now suppose i habe f of x to x to the n if you know were math people gotta always do it, ones to much got to go to the end so its going to be the limit h goes to 0 |

5:31 | x plus h to the n minus x to the n over h you multiply that out you get something thats called a binomial expansion which most of you learned in high school and probably forgot but some of you its stuck in there okay the binomial expansion is what you get when you multiply out a whole bunch of h's you get x plus h times x plus h |

6:00 | times x plus h times x plus h n times so you just keep multiplying those so whats going to happen, well the first term youre gonna have x times x times x n times were gonna have x to the n sign |

6:31 | so that term will cancel with the x to the n now we have an x to the n term were gonna have x to the n minus 1 and n minus 2 and so on, the next term we have is the x to the n minus 1, you take my word for it that occurs n times i dont know if you can read that but ill fix that |

7:08 | so thats what your second term will end up being if you multiply this out all of the remaining term will have more than 1 h when you multiply this out you will no terms power of h youll have a term thats 1, youll have a term thats 2h and a term that has 3, and so on until you have a term that you get n h's |

7:31 | and similiarly youll have a term h to the n x to the n-1 x to the n-2 and so on, thats what the binomial expansion does you can get that out on your spare time so all of the remaining terms here will contain higher powers of n of h, theyll have h squared so when you pull the h out |

8:00 | were gonna get something like this were gonna have h and then x to the n minus 1 and then well have a bunch of terms that all include h's so when i do the limit all the terms of h are 0 and the only thing left standing is nx n minus 1 |

8:31 | so this gives us a rule for derivatives that your friends have been showing off now you can show off too do his at parties do this on dates i tend to not get 2 dates, one is all they can stand you know i advise you no calculus on the first date |

9:02 | if f of x is x to the n then the derivative n x to the n minus 1 that is called power rule by the way the next two or three weeks of calculus still have a whole bunch of rules must of which i would not actually devide |

9:32 | i sort of derive so those are pretty letter what does that mean well lets do some f of x is f to the 5 ready the derivative |

10:01 | so lets us our formula n to the x minus 1 so its the power you put the power in front and decrease the power by 1 and thats the derivative so instead of all that hard work you just do that you could do it where the h goes to 0 but why theres a reason why we invent these short cuts because we dont want to do it the wrong way either, you should understand how it works and where it comes from but at some you say alright and show me the fast way |

10:32 | thats the derivative if i say f of x is x to the 20, f prime of x is 20 x to the 19 this is not hard, so you also subtract 1 people who are god at subtracting 1 |

11:14 | now of course you all can subtract 1 right so now i say whats the derivative of the square root of x i know you know but you took calculus in high school anyone else took calc in high school yea you dont have to raise your hand we know who you are |

11:40 | so how owuld you take the derivative of that well first make that x to the 1/2 okay ready so the derivative is 1.2 okay subtract one for 1/2 |

12:02 | negative 1/2 you wpuld be amaze at how many of you will mess up subtracting 1 f of x is 1/cubed root of x you subtracted one so you know what to do now no calculator |

12:33 | well first we have to rewrite that so cubed root is to 1/3 so 1 over cubed root is 1minus 1/3 f prime of x is minus 1/3, what is minus 1/3 minus 1 not negative 3/3 |

13:03 | you all cant subtract one see this is what i meant minus 4/3 anytime you want to subtrcat 1 from a fraction you just you take away the denominator, you have to take away 3/3 minus 1/3 minus 3/3 is minus 4/3 so far so good |

13:31 | ill get to that dont worry, you didnt hear his question dont worry so thats the power rule now we can play with the power rule a little bit more for more complicated derivatives, were not sticking to this, to easy so some stuff you should know what if you have f of x equals 5 theres no x there is there what does the graph of f of x look like, f of x looks like |

14:04 | gonna look like that at 5, so whats the slope of that 0, so if we do f of x equals a conjugate, the derivative is 0 so lets add to that rule actually i take that back I put it over here when we wont limit them f of x is x to the n |

14:32 | f prime of x is n x n-1 f of x is equals the constant the derivative is 0 you also could of tholught of this if you wanted is 5 x to the 0, anything to the 0 is 1 and then when you bring the 0 in front its not really valid but if it helps you, you can do that |

15:01 | what if we just have f of x equals 11x well thats the line, thats the line y equals 11x whats the slope of y equals 11x the derivative of the is 11 |

15:30 | cause this is 11x to the 1 you bring the 1 down you get 1 11x to the 0 and x to the is 1 if you get f of x and the constant times x the derivative is the constant so far so good now one more what happens if we have a contant times x to the x |

16:00 | if we call a k in there what if f of x is 10x to the 7 you put the 7 in front and multiply it by 10 i can prove this or you can just take my word for it so its 70 x to the 6 |

16:32 | so these are all kind of related this is kx to the n the derivative is k tinmes nx to the n-1 next monday were gonna review these and teach you a couple cases all sorts of fun stuff |

17:00 | so lets practice with this for a couple minutes and then were gonna do some exam practice isnt this fund, much easier than doing the whole limit well of course thats why we evented calculus if we had to do it the hard way you wouldnt do it |

17:30 | so what do think happens if we add a couple of these together suppose i wanted to find the derivative of that well you just add them together and subtract them again i can show you thats valid but just trust me so you just do one in front the derivative is 32 x to the 3 cause 8 times 4 is 32 |

18:05 | plus 7 times 5 is 35 x to the 4 3 times 2 is 6 subtracting 1, i told you its not hard except with fraction or with pi well its really pi minus 1 |

18:30 | how about that looks like fun im just making up numbers as you would see the hard part of calculus isnt taking the derivative its what you do with it thats hard the derivative its self is pretty straight forward |

19:05 | the derivative is 18x squared plus 16x minus 5 the derivative of that is 0 we can ask you for the second derivative you just do it agian take the derivative of the derivative and the 3rd and 4th and so on, some point you run out of derivatives and you just get 0 i can tell exactly whats happening at that point but ill save that for next week |

19:33 | that would be something to keep you in suspense now how will you know to do it this way or you have to do it the hard way well we might say use the definition of the derivative then you would have to do it the long way but now youll know what youre looking for so youll know if you messed up thats like when i do it on the board and stop and go oh wait i made a mistake okay |

20:00 | if we just say find the derivative so lets do a sample question |

20:52 | something like this we give you and equation for f we say find f of 2 find f of prime of 2, by doing the derivative, did i say whats the definition of the derivative |

21:06 | then you dont have to, find the equation of the tangent line that equals 2, alright everybody lets see you do it i did this the other day, we just have to use the equation of the line to find the equation of the tangent line we need 3 things okay write that out the equation of a tangent line or any line |

21:31 | is y minus y1 equals m times x minus x1 you need x1 you need y1 and you need the slope you find the slope by doing the derivative part b |

22:03 | you can do the y-mx+b if you want yull just make it hard on yourself do the minimal amount of work on these questions okay |

23:42 | alright thats long enough lets find f of 2, f of 2 is easy thats only not calculus thats some where around 9th grade f of 2 you plug in 2 f of 5 times 2 to the 3 minus 7 times 2 |

24:00 | plus 2 40 minus 14 plus 2 equals 28 so 28 thats a that that is going to be y1 that is your y coordinate that goes with you x coordinate 2 so now all we need to do is find the slope so f prime of x |

24:32 | put the 3 in front and you get 15x squared 7x is just 7 the derivative of 2 is 0 f prime of 2 is 15 times 2 squared minus 7 which is 53 so far so good alright so the equation is y minus 28 |

25:01 | s 53 times x minus 2 thats it easier than the mx+b thing alright should we try another one of these |

25:31 | you should be able to do the equation of a tangent line, lets do one more anyone taken trigonometry |

26:36 | remember theres no calculator so practice without a calculator how do you know 53 is m, the derivativ gives you the slope of a tangent line m is the slope of f okay so thats how you know |

27:27 | if f of x is 6x to the fourth plus 3x squared |

27:31 | find the equation of the secant line through 0,f0 and 2, f of 2 find f prime of 2 again i would hae no trouble with that c. find the eqaution of the tangent line at 2, f of 2 find the equation of the secant line so how do you find the equation of the secant line |

28:00 | oh i actually meant the slope of the secant line im sorry but we can do the equation not that hard the slope well it would be f of 2 minus f of 0 minus 0 f of 2 is 6 times 16 is 96 |

28:30 | plus 3 times 2 squared thats 96 plus 12 108 f of 0 is 0 so the slope is 108-0/2-0 54 |

29:06 | well this 108 minus 0 over 2-0 the slope is 54 so the equation would be y-0 |

29:32 | equals 54(x-0) or so that would be the equation of the secant line now lets find f prime at 2 well f prime is x is 6 times 4 is 24 x cubed 3 times 2 is 6 6x plug in 2 |

30:04 | 24 times 2 cubed plus 6 times 2 i dont think we give you numbers this scary 24 times 8 is 192 you can simplify that okay so thats f prime of 2 |

30:30 | and now the equation of the tangent line you could do y minus f of 2 which is 180 is 204 times x minus 2 alright other stuff you should be able to do |

31:16 | lets do some graphing i was thinking today where would the trouble sbe on the exam what are the trouble spots going to be drawing the graphs and interpting the graphs everybody feel comfortable with this cause im gonna erase this in a second |

31:33 | i can erase awaya well originally i was asking for the slope but find the equation of the secant line except you do the slope of the secant line which you can find by 2 points f of 2 minus 2-o correct you just use the original function and ou evaluating at 2 points remember the derivative is when youre shrinking those 2 points |

32:01 | infinitely close to eachother the secant line is evaluating two points the tangent line is at 1 point 2 points at distance h apart but h turns to 0 so its only one point the derivative of the tangent line is when you find the derivative of a single point what we do is we make one point magically 0 so were doing 2 point 1 point method |

32:39 | so lets practice some graphing from there |

33:05 | alright you want to remember whats going on with graphing the derivative and all this stuff you have a graph lets say that x and f of x |

33:30 | and we ask you to graph the derivative okay so were gonna graph the derivative so were gonna do this as a team a team means i do it and you take notes, so lets think about whats going on here |

34:09 | remmeber derivative is slope so whats the slope here imagine youre on the graph going down going down pretty fast to the negative slope were going down and wait were bottoming out so at that minimum spot |

34:30 | well call that 2 slope is 0 before there we just had negative numbers so our derivative would have a negative value so if were graphing it, it would be below the x axis something like that it could have an under curve doesnt really matter |

35:02 | now what happens from here up, now we start to go up thw graph and some point 0 again the derivative is 0 here and the derivative of 0 is here which well call 4 |

35:32 | and the meantime it has positive values it has positive values, it gets steep for a while and then it turns around and goes back to 0 so it has to come back dow positive value and then at some point it has to come back to this and then here they have negative values again and they just get more negative so something like that, or make i t a little prettier |

36:11 | so that graph looks like that graph so by the way thats a negative x cubed thats a negative x squared the derivative of x3 s 3x squared so notice it goes from the cubic graph to a quadratic straight forward good lets make sure you could do one |

36:35 | so much easier when i just do it and you watch thats x and f of x |

37:00 | draw the derivative graph what would the derivative graph look like well heres a clue when you do derivative graphs or not clue a tip tip number 1 figure out where the maxima and minima are, the maximums minimums thats your first clue, that has a maximum right here a minimum |

37:31 | right here and a maximum right there there called relative maximums which mean in the area its the biggest value the slope is 0 at a maximum or minimum because your going up its 0 then your coming down again its 0 then you go up then its 0 and comes down so we know where the 0s are 0 at -3 0 at -2 and 0 at positie 3 |

38:03 | you just look at the sign of the derivative the derivative is going up here the derivative is positive the curve is going up the curve is increasing the curve is positive so you just need positive values than at -3 and -2 the curve is going down so the derivative is negative until you get to -2 again when you get bacl to 0 |

38:31 | something like that and it could be deeper doesnt matter just a sketch then between 2 and 3 the curve is going up til it gets to 0 and then its going down there you go something like that by the way that x to the 4 that is -4x cubed or something some various on that |

39:02 | were gonna do one of thpse going in the other direction is harder notice what happen at minima and maxima adding minimum the slope is negative the 0 and then its positive and the maximum the slope is positive then 0 then its negative, i wrote this down before but lets do it again |

39:35 | slope is negative 0 positive so the left side of where the minimum is the slope is negative the right side where the minimum the slope is positive and at the minimum is 0 because its going down to 0 and then uo thats not hard and at a maximum its the other way around |

40:03 | whats the slope doing well its going up to positive then it stops then the curve goes down again so now if i gave you the graph of the derivative and i said where would i have a maximum and minimum you would tell me |

40:51 | suppose i give you this now this is the derivative graph |

41:01 | so again consantrate on the 0, so -4 -2, 2, 4 so notice you get 0 right here at negative 4 to the left of 0 the slope is positive at a positive value dont look at the direction the graph is going just notice its above the x axis then on the right side its below the x axis negative so its going positive negative negative |

41:32 | thats a maximum so you have a maximum at negative 4 just gonna write maximum now at negative 2 were below the x axis then 0 the above the x axis so its negative 0 positive okay at 2 were go from a positive value |

42:01 | to 0 to a negative value so thats a maximum again there going to alternate theres always a therom about that and at 4 were below the axis negative\ 0 positive so thats a minimum now with out any information you have no idea where this graph is really located doesnt really matter you graph the maximum |

42:31 | thats a minimum thats a maximum minimum good enough ] so we gave you an asortment of graphs but what were ooking for has to look something like that it could be located anywhere we would give you another key point nformation if we really wanted you to pin point the graph so we would give you info like f of 0 is 5 so thats good if we said f is negative the graph would be down here we just want sketch we just want you to show where tha maximum and minimus are |

43:08 | we can do one more oyu guys love this stuff okay how about |

43:40 | alright what if thats the derivative graph what would the regular graph look like you have a 0 here |

44:00 | a 0 here, at the origin so notice positive values 0 negative values so we have some kind of maximum right there and at the origin we have negative values 0 positive values so we get some kind of minimum right here now to the left to the left of this spot we have only positive values but they kind of go up its steep and they kind of go back down to 0 |

44:34 | so its going to be 0ish something like that, lets see if that makes sense 0 going up and then yea it could be more dramatic than hat but it doesnt matter then we have negative values and get back to 0 |

45:02 | thats a minimum then we have positive values and we get 0 again so we have 0 ish |

45:41 | this just tells you the values of the slope and if here positive or negative numbers tell you the location of the graph the curve is going up and the curve is going down nd then going up, its to close to 0 it isnt do much |