Stony Brook MAT 125 Spring 2015
Lecture 11: The chain rule
March 9, 2015

Start   we have a bunch of rules that we have learned now we learned the power rule this is how you find the derivative of something raised to a power y=k times x to the n derivative k times n x to the n minus 1
0:30for example you have y equals 20x to the fofth take the derivative 100 x to the fourth because its 20 times 5x tothe 4th so you bring the 5 in front 5 times 20 is 100 x to the fourth if you wanted to do the second derivative would be 400 x cubed, if you wanted to do the 3rd derivative
1:05it would 1200x squared if you wanted to do the fourth derivative it would be 2400x the fifth deivative i dont know when ill use the fifth derivative but it would be 2400 so notice what the 6th derivative would be so the 6th derivative would be 0, so if you want read it
1:31its 20 times 5x to the 4th becomes 100x to the 4th 400x to the 3rd 1200x squared 2400x 2400 and the derivative of the constant is 0 so if y is a constant the derivative is 0 everybody remmeber this
2:06if y is a constant times x the derivative is just the constan so when we had 2400x the derivative is just 2400 the derivative of 2400 is just 0, so notice the 5th derivative 20x tot he 5 so if we took the 6th derivative youd get 0 so if you have a function raised to the n the n plus 1th derivative is 0
2:30so if you have y equals x to the thousan the thousand and first derivative is 0 the milion derivative would be 0 so once past a thousand the derivative is 0 one more time so if y y is x to the 20th then the 20th derivative
3:00the 20th derivative instead of y prime just write this 20th derivative would be 20 factoral showed you that last time and the 21st derivative is 0 do you remmeber why its 20 factoral keep bringing the numbers in front so its down one evertime
3:30alright so notice you have 5 then 4 then 3 then 2 then 1 so each time youre multiplying so you get 5 multiplied by 4 times 3 times 2 times 1 thats 5 factorial you have n it becomes n factorial y is x to the n the derivative the nth derivative the n derivative is n factorial and n plus 1
4:000 because the derivative of any constan is 0 so notice ive been using these 2 notations, the prime notation and theres the d d letter, d x notation this is the notation you should get more comfortable with even though you like the prime because its the short hand theres reasons its better to use the other notation if you do other types of problems you pretty much have to be able to use this type of notation
4:33so thats the power rule then you have the product rule the product rule is when you have 2 functions multiplied together the derivative is the first times the derivative of the second plus the second times the derivative of the first or the other way around
5:01i know you had a few webassign problems that look like this but for example if you had say i have xcosx first derivative now theres to functions, the function x
5:30and theres the function cosx the first function times the derivative of the second plus the second function times the derivative of the first whats the derivative of that 1, okay 1x so its just the letter dy/dx of x is just 1 because thats this rule
6:06y is the constant times x the derivative is just the constant so if this was 5xcosx then that would just be a 5 this is annoying but not so bad, what if we wanted to find the second derivative this would require the product rule
6:33the minus xsinx term so the derivative of that would be minus x times derivative of sin which is cos plus sinx times the derivative of -sinx which is 1 the derivative of cos is negative sinx
7:02that is minus xcosx minus sinx minus sinx minus xcosx minus2 sinx, why did we do that because we did another derivative and set it equal to 0 you want to get it as simple as possible beofore you take the derivative agin so thats the product rule we didnt do, the triple product rule
7:33so suppose you had wed do something like that, x squared sinx secant x now we have 3 things so how do you do there derivative when you have three things
8:06so you can do you can pretend its two functins like that and the derivative of this requires the product rule we do the derivatove ots the first function times the derivative of the second, x squared sinx times the derivative of secx you know whatt eh derivative of sec
8:34secx tanx lus secx now you do the derivative of x squared sinx you need the product rule you do x squared times the derivative of the sec plus secx times the derivative of sinx
9:01that requires the product rule which is x squared tims the derivative of sin plus sin times derivative of sinx, 2x okay notice you get three terms you get this term this term and sec times this term times sinx if you do the derivative of that a second term thats pretty nasty
9:39but if you have f times g times h and if you were doing the derivative would be f of x g of x times the derivative of h
10:01plus h of x times the derivative of f and g which is f of x g prime of x plus g of x f prime of x notice what happens if you simplify tht out you get f of x g of x h prime of x plus f of x h of x
10:31g prime of x plus h of x g of x f prime which it kind of makes sense so its leave the first two functions alone itimes the derivative of the 3rd plus the first times the 3rd times the derivative of the second plus the second times the 3rd times the first so two of them and the derivative of the third one then you rotate it 3 times instead of 2
11:08so thats all the product rule stuff
11:44cn i repeat the equation sure you have f times g times h the derivative is f times g times h prime times f pluss h times g prime plus g times h time f prime
12:00you do two of them times the third, then you switch it and switch it again so each one take the derivative and multiply it by 2 non derivatives
12:47quotient rule quotient rule is similar to the product rule except in the product rule it doesnt really matter
13:03if you do f times g prime plus g time f prime or the other way around its the first derivative 2nd plus second times 1st or you can just switch it but for the quotient rule order matters because you are doing subtraction in addition it doesnt matter what you do 3+2 is the same as 2+# but subtraction is not the same 3-2 is no the same 2-3
13:32you discovered that somewhere in fourth grade so in qoutient rule y=f of x over g of x the derivative of g f of x times f prime of x minus f of x times g prime of x over g of x squared
14:08so there the order matters for example if you had
14:32if you had 3x squared plus 5 pver sinx i cant imagine when youd actually use a problem like this derivative so lidehi, hidelo lodehi-hidelo over lo squared
15:00so lo is the lo function and high is the high function lodehi so lol times the derivative of the high function just 6x minus reverse 3x squared plus 5 times derivative of sin which is cos over bottom squared and you only simplify that if you really have to we never really want to simplify we just leave it alone
15:32in fact if we ask you to plug in a number so if we ask you to evaluate this thing pi/6 you immediately plug in you do not simplify its very easy to simplify theres numbers, simplify aromatic then simplify algebra remember that as a tip whens day were gonna do hard stuff and plug in if we said find this, so here dy/dx at x equals pi/6 by the way i know theres some confusion
16:04on webassign this notation bar even the little x equals pi/6 means evaluate this at x equals pi/2 thats what that notation means if we said x equals 1 wed get a value of 1 and so on it would just be an awful number this would just be sin pi/6 which you all memorized is a 1/2
16:30times 6 times pi/6 which is just pi this would be 3 pi/6 squared plus 5 cousin pi/6 all over sin of pi/6 squared cospi/6 is 3 radical 2 right this simplifys somewhat
17:02sin of pi/6 is a half 6pi/6 is pi
18:00thats quotient rule some other stuff then you get to the new stuff so we have the power rule we got the product rule we got the qoutient rule
18:45so derivative of e to the x is e to the x im going to do this with primes this time cause im lazy so whats the second derivative of e to the x e to the x then the third derivatove e to the x
19:02remember that its just e to the x all the way to the end the nth derivatie of e to the x did i do natural log of x yet no thats coming then theres sin and cos, the derivative of sin is cos were gonna go with cos negative sin
19:30now remember when you have a trig function you have the derivative when you want to do the derivative of the cofunction its just a negative of the co of the original function we have y is tangent the derivative is sec squared o were gonna do the co derivative, cot
20:01now we can switch this is the co function the co function sec squared, csc negative csc squared and if i had sec derivative of secx tanx
20:32and then for csc thats a cofunction youre gonna do csc and cot
21:32suppose we have y equals sinx the derivative is cosx seond derivative is negative sinx the third derivative is negative cosx the fourth derivative is back to sinx so if i do the fifth derivative its going to be the same thing as the first derivative
22:00if i do the 6th deriative and so on, its going to be a cycle then its just going to be cos minus sin min cos and so on kind of the way i did, theres a relationship in there if you wanted to find the nth derivative of sinx you would do you would take it divide it by 4 and keep the remainder
22:34if you got a remainder of 0 it would be sinx that would be remainder 0 remainder equals 1 remainder equals 2 remainder equals 3 say i wanted to the 101 derivative if i have y is sinx find the 101 derivative
23:02of sinx 4 goes into 101 25 remainder of 1 so that would be the same as the first derivative cosx if i wanted to do the 232 234 derivative
23:37well 4 goes into 234 58 times with 2 left over so how do we say we get the second derivative second derivative was minus 1 okay theres a cycle for this the cycle for e to the x is 1, its just the same thing over and over again
24:00with sin and cos with every 4th it goes back to where it started there a relationship in there which so far so good, alright lets learn something new chain rule
24:33this is where people start to have trouble in calculus now professor sutherland should of put up a video about this lets see if its true videos have to go to our math page maybe not yet but i think he is but im wrong a lot
25:01just in cas i think he is but im not going into detail today msybe itll be up on wednesday i know youre excited the chain rule is what you do when you have a function thats inside another function back in precalc when we do all that f of g and g of f this is y i know you were wondering this is why how do you do the deriative of a function inside another function
25:31well what you do is do the derivative of the outside function you leave the inside function alone times the derivative of the inside function you look at that and you say that doesn look so bad what does that mean?
26:04you have y=(5x squared +3x+1) to the fourth you want to find the derivative now you certainly certainly could just multiply this out it would take a lot of time but you'd certainly get there or you do the chain rule chain rule says you have two things going on here something to the fourth thats your outter function
26:32and then you have whats inside you have two things, the outside function to the fourth and the inside function, you want to do the derivative of the outside you just do 4 times whatevers on the inside to the 3rd leave the inside alone this just stays 5x squared plus 3x plus 1 times the derivative of the inside which is 10x plus 3
27:04that wasnt so hard was it okay lets do another one suppose we had y=sin(pi/x) so we have an outter function sin then we have an iside function pi times x
27:30outside function is sin of inside function is pix so the derivative of the outside function is just cos leave the inside alone times the derivative of the inside which is pi thats it, thats not so bad of course i can make it worse do a couple more examples so everybody can get the idea
28:13suppose i have y=square root tanx so the square root is to the 1/2 so this is the same this as writing tanx to the 1/2
28:32so if i wanted to do the derivative of that whats my outside function, my outter function the thing to a 1/2 so the derivative of that would be 1/2 whatevers on the inside -1/2 so then you can just fill in the inside you dont do anuthing for that
29:01times the derivative of the inside sec squared kind of like the analogy of a onion your kind of peeling away the layers of the onion, you go the outside the next layer and the next layer and you keep going in you can have multiple layers lets just practice one more
29:37why doesnt everyone take a second and do that dy/dx you have an outer function something raised to the 10, so 10 times the something tot he 9 so whats inside is x cubed plus e to the x plus 4
30:00times the derivative of the inside times 3x squared plus e to the x 10 x cubed plus e to the x plus 4 times 3x squared plus e to the x now ill make thi harder suppose i had y= sec squared pi over 4x
30:35so whats going on here we have 3 layers this is sec of pi/4x squared so you ask yourself what is happening to x first you multiply by pi/4 then you take the sec of that and then you square that 3 layers not 2 so if you want to do the derivative
31:042 leave the inside alone sec pi/4x raised to 1 times the derivative of sec pi/4x which is sec pi/4x tan pi/4x times the derivative of pi/4x
31:30which is pi/4 so you have 3 layers first you have 2 times the derivative of something to the 1 then you have the derivative of the sec which is tan then you have the derivative of the inside which ispi/4 this is the kind of question people put on the test and lots of people get it wrong you can see how easy it is to mess that up
32:10lets do another one make sure you guys get the idea okay why doesnt everyone try that
32:33how do we write this this is sin of x squared plus 1 all to the 1/2 again you say what am i doing with a half, well first you square it and have 1 then i take the sin of it then i raise it to the half the other most function is a half power so dy/dx is 1/2 the whole thing inside is raised to minus 1/2
33:05where this is left alone next i have to do the derivative of sin which is cos of x squared plus 1 so notice again i dont do anything to the x squared plus 1 because its the inner most of the function so i have 3 layers i have a 1/2 power i have sin and x squared plus 1 so first
33:30i just do a 1/2 inside to minus 1/2 then i do the derivative of sin which is cos of x squared plus 1 then i multiply it by 2x so theres 3 layer and of course you can keep going
34:06suppose you have that so again we have 3 things going on we have e raised to the sin of something and the something so the derivative well first we have to take the derivative of the outer most function which is e to the power so thats e to the sin4x-2 so notice you didnt have to do much with that
34:31it could just be the derivative of e to the x e to the x, derivative of e to the whatever is just e to the whatever now times the derivative of the sin which is cos of 4x minus 2 times the derivative of 4x-2 okay so 3 layers there e to the sin of 4x-2 times cos4x02
35:01times 4 how we doing so far were getting the hang of this? we can do a couple more we should, this is the thing youll have the most trouble with trust me
35:47there you go, product rule, chain rule you thought you were done of course you can simplify this first you can multiply this out
36:02but i could just make it harder the point is not to do that so we have to do the product rule and then in the product rule so if were doing derivatives thats going to involve chain rule product rule, first function times the derivative of the second function so the first function e to the 3x plus 2 plus x and now we want to do the derivative of the second fucntion
36:31what is the derivative of e to the 5x-4 well its e to the 5x-4 times the derivative of 5x-4 which is 5 plus 2x\ you have to do the derivative of e to the 5x -4 and e to the 5x-4 times the derivative of the power which is 5 theres 2 layers here e to the 5x-4, so the outter function e
37:04inner function 5x-4 the derivative of that is the derivative of e, the outside function 5x-4 times the derivative of 5x-4 which is 5 then you have the derivative of x squared which is +2x now thats the first half of the product rule plus got to reverse this e to the 5x-4 plus x squared plus 1
37:30times the deriative of this whats the derivative of e tot eh 3x plus 2 its e to the 3x plus 2 times derivative of 3x plus 2 which is 3 plus 1 the 1 comes from the x so in the chain rule notice if you have something thats y=e to the kx
38:02the derivative is justgoing to be e to the kx times k thats your chain rule e to the kx and the derivative which is k if you have y is sin to the kx sin of kx the derivative is cos kx
38:31times k so ou write it in front just so its easier to keep track of notice i get sin4x+2 cos4x-2 times 4
39:24suppose you had something like that so what do we need, we have both of the cordinates
39:32so we just need the slope remmeber this is calculus class the only thing you need to do is take the derivative so whatever else goes on take the derivative so whats the derivative of sin of 4x well i just showed you its cos4x times derivative of 4x which is 4 i can put the 4 on the other side the derivative of sin4x is cos4x
40:00times the derivative of the inside which is 4 this rule i just wrote down now we have to evaluate this at pi/8 dy/dx at x equals pi/8 is 4 cos 4pi/8 is pi/2 whats cos pi/2 0 cause you guys all know the circle
40:30this is really easy because i didnt really pay attention when i made the example y-1/2 0 times who cares x-pi/8 so its just y-1/2 0 couldve done a better example but thats okay we can do one more
41:47if the position of a object is given by x of t equals tsi(3t) find the velocity and the acceleration so for those of you who dont know the velocity is the first derivative and the acceleration
42:03is the second derivative if you havent taken physics is x of t is t sin 3t find the first and second derivative lets do the derivatives so the first derivatives derivative 3sin3t so its t and how do we do the derivative of sin of 3t
42:32cos 3t times 3, you can out that in front, plus the derivative of t which is just 1 times sin of 3t or if we want to make this a little easier to look at its 3t cos 3t plus sin3t
43:00okay howd we do second derivative so t of t is the first derivative a of t is the second derivative okay were going to need the chain rule to do this one so 3t the derivative of cos of 3t is minus sin of 3 of t times 3 plus the derivative of 3t which is 3
43:33cos 3t plus the derivative of the sin piece which is 3 cos 3t alright again its 3t times the derivative of cos 3t which is minus sin 3t times 3 plus the derivative of 3t which is 3 cos 3t
44:00plus the derivative of sin of 3t whihc is 3cos3t and of course youd simplify that if you had to go one more round which is -9t sin of 3t plus 6cos3t so the first derivative is velocity and the second derivative is acceleration you want to know what the 3 derivative is its change in acceleration
44:36alright i think everyone had enough for one day