Stony Brook MAT 125 Spring 2015
Lecture 09: The product and quotient rules
March 2, 2015

Start   this piece of the course between now and certainly before spring break and a little after spring break were just going to focus on techniques of differentiations how you take derivative of things so youve all learned the power rule but were going to review that for a minute thats how you take the derivative of something to the power and then of course you have to learn to do the derivative of all sorts of things and in a few weeks youll be able to take the derivative of anything and then later in life youll be able to do derivatives even if nothing else so remember the power rule
0:33the power rule says that if y is k times x to the n the derivative of y k times n -1 n-1, x
1:07nx to the n-1 and notice you can make this f of x and f prime of x you can do y you can do y to the x this is generally a better notation its not as compact but as with calculus i can be whats called a live bit notation e y to the x notation better one to use later it becomes clear
1:33for example if y=7x to the fifth dy dx 35x to the 4 cause i bring the 7 down and i multiply 7 by the 5 x reduce the power by 1 this is very straight forward right and for polynomials you have a whole string of these add and subtract them
2:02you can just do them all together so if yu have y=6x to the 4 -4x cubed plus 11x squared minus 8x plus 10 you take the derivative and these you should generally be able to do in your head
2:32you bring the 4 down 6 times 4 is 24 24x cubed you brin the 3 down 3 times 4 is 12 reduce the power by 1 12x squared 11x well the power here is 1 so its 11 to the 0x which is 11 thats suppose to be a squared 11x sqaured sorry if you wrote it in pen
3:00thats 22x derivative of 8x is 8 the derivative of 10 is 0 so lets learn about some of these special ones that will be very useful y=kx derivative just the constant so when you have y its -8x the derivative is -8\ do not confuse this with y thats the constant
3:34then the derivative is 0 here the derivative of 10 is 0 cause 10 isnt changing the derivative tells me how much something is changing so whats the rate of change of 10 well the rate of change doesnt change cause its 0 now lets work on a couple special cases
4:00we love square roots not sure why but they show up a lot and thats one place people tend to mess up so how would we take the derivative of the square root of x well you could rewrite it as x to the 1/2 then you take the derivative bring the half in front subtract 1 and you get 1/2x to the minus 1/2
4:30this simplifies rather nicely to 1/2 radiacl x so i recommend that you memorize this one its gonna become very useful if you do, y is the square root of x derivative 1/2 radical x heres what happens other words every time you have the square root
5:01youll write lets see 1/2, 1/2 youll have trouble with the algebra because you will not be processing this as the same as this so whats gonna happen in a while youre gonna get the derivative and set it to 0 or something and this is not as easy to solve as this is you have to memorize you guys like to memorizes thats hat biology
5:34another one, oh i walked away what if y is 1/x well if y=1/2 we can write that as x to the /1 the derivative is negative 1x to the -1 which we can write as
6:01-1/x squared s again at the square root of x youll find it very useful if you memorize trust me when i tell you life will be better if you memorize those 2
6:31so far this isnt very hard right?
just rules were not even making you learn where the rules come from i kind of did that with the power rule last week you can go back and look at the video thats the miracle of video tthis is kind of interesting learnig these so lets learn more interesting functions
7:31heres a really tricky one y=e to the x the derivative is e to the x i think professor sutherland proved it in his video i will be honest i have not yet seen it but ill watch it tonight lets do an example
8:05y= e to the x derivative e to the x pretty hard right what if y is 5e to the x the derivative is 5e to the x so e to the x is great i can do anything to it its indestructable its just e to the x
8:34and we basically needed a function thats itself so we invented one called e to the x its not the only reason its true but its a good reason why its true e to the x shows up everywhere in biology and chemistry so it has to deal with the way things grow and decay so you see it a lot and the rate its changing is always related to the way its changing i know it sounds silly but thats why the derivative is always the same
9:51product rule
10:13the product rule is what you do when you have two functions multiplied together so you have function number 1 times function number 2 so what do we do
10:31well your probably just saying the derivative of funtion 1 times he derivative of function 2 wouldnt that be nice if all you had to do was take this derivative of this times the derivative of that but unfortunately its not that simple so you take the first one and multiply it by the derivative of the second one and then you reverse
11:08weird but theres weirder things lets do an example suppose we have y=
11:31x squared plus 3x times 5x cubed minus 2 and you have to take the derivative of that now you can certainly just foil write it out and find the derivative and if you have a pair of binomials like this thats not a bad way to find the derivative its pretty quick but of course we can give you messier functions throw in a pi maybe a tangent something like that
12:09but the rule is you take the first function leave it alone and multiply it by the derivative of the second fucntion the derivative of the second function bring the 3 down you get 15x squared minus 0 and now we reverse
12:32plus the second function times the derivative of the first function and yur done you are not expected to simplify feel free to simplify if you want why would you want to simplify because if you have to do the derivative again if you have to solve for 0
13:00or something like that otherwise thats it your done thats not so bad and by the way in the product ruel since your adding the order doesnt matter you can do it the other way by the way lets just multiply that out and see what happens you get 15x to the 4 plus 45x cubed plus 10x to the 4 plus 15x cubed
13:31minus 4x minus 6 which is 25 x to the 4th plus 60x cubed minus 4x -6 now why did it do that, lets it the other way, multiply it out
14:045x to the 5+15x to the 4 minus 2x squaredd minus 6x and now take the derivative and see if it comes out the same and it does you get dy/dx 5 times 5 is 25 so 25x to the 4 4 times 5 is 60
14:33minus 4x minus 6 there the same so either way works so what would i want to do it this way when i can do it this way these could be more complex functions there is no advantage of doing it this way verses this way it really depends what you want to do next but just with a pair of binomials it doeant really matter
15:03it could be 6 terms then you really want to use the proper rule alright lets do another example to make sure we go the idea find the derivative of the product rule
15:32so the derivative take the first function leave it alone multiply it by the derivative of the second function so 5 times 4 is 20 reduce the power by 1 7 times 2 is 14 reduce the power by 1, plus 3
16:00plus if we run out of room i will write it underneath i was just gonna say when you do product rule give yourself a lot of room especially if you write large now reverse plus 5x tot he 4 plus 7x squared plus 3x times the derivative of this which is 12x squared minus 12x
16:31plus 0 we would not expect you to put that mess together more likely we would say something liek evaluate that at x equals 1 even x=2 starts to get annoying but you should be able to do x=1 because x equals 1 lets do one more of these to make sure you guys
17:01know what you are doing
17:35alright find the derivative of tht lets do this one the derivative you leave the left term alone times the derivative of the right hand term the derivative of the right hand term is e to the x
18:01x squared is 2x now reverse e to the x minus x squared now the fun one, x+ square root of x plus cube root of x so 1 square root of x is now memorized so thats 2 to the x unfortunately the cube root of x i mean i know it but this is x to the 1/3
18:32so you should write it as 1/3 x to the minus 2/3 and leave it alone from there get comfortable with those exponents what is x to the minus 2/3 that is 1 /3 times 1/ cubed root of x squared thats what x to the -2/3 means if your not sure on all this fractional exponent stuff
19:01review it it will show up a lot especially when we start doin word problems the fun part of the course after the second midterm by the way i already scheduled the review session sunday before 5 oclock in this room so much fun so this is the product rule why is the product rule what it is
19:30well ill give you a little bit explanation so remember derivative is change in something so suppose you have a rectangle and right now we know th area of the rectangle the sides are x y we know the area of a rectangle is xy so the derivative of how something is changing so lets say x gets a little bigger
20:00and y gets a little bigger what happens to the area so if x gets a little bigger by lets say e to the x sorry this way dx and y gets a little bigger by dy now our new part of the rectangle the length of the rectangle is now y+dy
20:36and x+dx so thats our new length with d as a tiny amount but remember in calculus you are always shrinky to 0 but we want to know how its changing so the new area is now x plus ex plus y times dy
21:02before the area was xy and now we added a little bit in the x directin and a little bit in the y direction x plus dx and y plus dy so how much additional area did we add if we take the area of the new rectangle and subtract the area of the odd rectangle and well get how much it changed and thats the derivative so lets foil this out so we get xy plus x y
21:33plus ydx plus dxdy you know what i mean by foil right?
first the outside the inside and the last so how much did it change well the old area was xy and the new area is this so if you subtract xy it changed by this amount which is x times dy plus y tomes dx plus dy
22:02dx now thats the product rule x times dy plus y time dx and then you have this product rule remember dy is very small like 1/ a million dx is very small 1/ a million so dydx is a tiny number times a tiny number which is a really tiny number so you cans ort of throw this away cause the error is so small
22:30is what the product rule looks like so thats sort of a graphical explanation why the product rule is the way it is you are changing y by a little but you change x by a little bit this is how they change all together ignore this little bit, and we left those little bit because there liek 0 so thats the product rule now were gonna learn the quotation rule
23:11we have y is a function divided by a finction were gonna use the quotation rule the quotient rule kinda like the product rule but a litle different of course
23:32the main different between the product rule and the quotient rule for memorization is in a product rule f time g prime a prime you can do it in any direction it doesnt matter quotient rule the order matters the derivative is the function on the bottom times the derivative of whats on top minus reverse the order the function on top times the derivative on the bottom
24:04over the bottom function squared later when we do chain rule youll understand where that comes from bottom derivative minus top bottom top minus derivative of the bottom or bottom squared nicely monitor this lodehi-hidelo
24:32lodehi-hidelo/lo squared did any of you look at the pages from my book i put up before the exam i hope they were some what helpful i would be putting up pages on power rule quotient rule chain rule product rule quotient ruke power rule this week so this is something you want to memorize again you have a binch of rules to memorize its sort of like were giving you some tools for you tool box
25:00you want to use all those tools properly so lets do an example suppose we have y= x squared plus 3x plus 1 over 5x to the 4 plus 2x cubes
25:36the derivative lidehi-hidelo so the lo function times the derivative of the hi function which is 2x plus 3 start to get comfortable with that x squared derivative of x squared is 2x derivative of 3x is 3
26:01derivative of 1 is 0 minus reverse top function hidelo times the derivative of the bottom function over bottom squared
26:33we defintely dont expect you to simplify these sometimes it pays to simplify these because youll have to set them equal to 0 or youll have to take another round of derivatives called the second derivative but for now you get something like that you just ignore it you just say fine stop right ther its our job to figure out if you did it correctly
27:38suppose we asked you to find the equation of the tangent line y= 2e to the x minus 1/ 2e to the x plus 1 at x equals 0 when you see the equation of a tangent line youre going to need to use y-y1 and m time x-x1
28:02we need to find y1 we have x1 thats 0 we need to find y1 and we need to find the slope slope of course is the derivative we are in calculus class first lets find y1 y would equal 2e to the 0 minus 1 over 2e to the 0 plus 1 well e to the 0 is 1 because anything to the 0 is 1 possible except with 0 and infinity
28:302 times 1 is 2 minus 11 is 1 2 plus 1 is 3 y1 is 1/3 thats a nice easy number now lets take the derivative dy/dx bottom function times derivative of the top function whic is e to the x minus the top fucntion times the derivative of the bottom function which is also e to the x
29:09all over 2 e to the x plus 1 squared now we evaluate that at 0 so remember e to the 0 is 1 at x equals 0 were going to get
29:312+1=3 2 times 1 is 2 minus 2 minus 1 is 1 times 2 over 3 squared 4/9 so you get y-1/3 4/9 x minus 0
30:07everybody understand that so the plugging in again even the 0 with 1 this just becomes 2 plus 1 is 3 times 2 minus 2 minus 1 is 1 times 2 over 2+1 is 3 squared
30:32you can not cancel this e to the x with this because you would need one here as well often however if you had to take the second derivative of this theres a lot o canceling because you get 2e to the x times 2e to the x here minus 2e to the x 2e to the x those would cancel you make your life a lot happier alright lets just do one more to get the idea
31:36alright lets just take the derivative of that alright derivative lidehi-hidelo over lo squared so lo derivative of the hogh one alright what do we do with 5 second root of x well
32:035 times defivative of square root of x which 1/2 to the x or you can make it 5/2 to the x minus 2x thats the first part minus 5 radical x minus x squared thats messy
32:34times the derivative of the bottom which is 3x 4 all over x cubed plus 4x plus 1 squared how did we do on that goo okay thats enough for today see everybody on wednesdy