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

0:30  for 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

1:05  it 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

1:31  its 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

2:06  if 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

2:30  so 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

3:00  the 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

3:30  alright 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

4:00  0 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

4:33  so 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

5:01  i 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

5:30  and 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

6:06  y 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

6:33  the 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

7:02  that 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

7:33  so
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

8:06  so 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

8:34  secx 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

9:01  that 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

9:39  but 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

10:01  plus 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

10:31  g 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

11:08  so thats all the product rule stuff

11:44  cn 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

12:00  you 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

12:47  quotient rule
quotient rule is similar to the product rule
except in the product rule it doesnt really matter

13:03  if 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 32 is no the same 23

13:32  you 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

14:08  so there the order matters
for example if you had

14:32  if you had 3x squared plus 5 pver sinx i cant imagine
when youd actually use a problem like this
derivative so lidehi, hidelo
lodehihidelo
over lo squared

15:00  so 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

15:32  in 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

16:04  on 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

16:30  times 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

17:02  sin of pi/6 is a half
6pi/6 is pi

18:00  thats 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

18:45  so 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

19:02  remember 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

19:30  now 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

20:01  now 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

20:32  and then for csc
thats a cofunction
youre gonna do csc and cot

21:32  suppose 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

22:00  if 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

22:34  if 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

23:02  of 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

23:37  well 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

24:00  with 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

24:33  this 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

25:01  just 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

25:31  well 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?

26:04  you 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

26:32  and 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

27:04  that 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

27:30  outside 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

28:13  suppose 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

28:32  so 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

29:01  times 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

29:37  why 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

30:00  times 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

30:35  so 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

31:04  2 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

31:30  which 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

32:10  lets do another one make sure you guys get the idea
okay why doesnt everyone try that

32:33  how 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

33:05  where 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

33:30  i 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

34:06  suppose 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 sin4x2 so notice
you didnt have to do much with that

34:31  it 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 4x2
okay so 3 layers there
e to the sin of 4x2
times cos4x02

35:01  times 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

35:47  there you go, product rule, chain rule
you thought you were done
of course you can simplify this first you can multiply this out

36:02  but 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

36:31  what is the derivative of e to the 5x4
well its e to the 5x4
times the derivative of 5x4 which is 5
plus 2x\
you have to do the derivative of e to the 5x
4 and e to the 5x4
times the derivative of the power which is 5
theres 2 layers here e to the 5x4, so the outter function e

37:04  inner function 5x4
the derivative of that is
the derivative of e, the outside function 5x4
times the derivative of
5x4 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 5x4
plus x squared plus 1

37:30  times 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

38:02  the 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

38:31  times k
so ou write it in front just so its easier to keep track of
notice i get sin4x+2
cos4x2 times 4

39:24  suppose you had something like that
so what do we need, we have both of the cordinates

39:32  so 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

40:00  times 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

40:30  this is really easy because i didnt really pay attention when i made the example
y1/2
0 times who cares
xpi/8
so its just y1/2
0
couldve done a better example but thats okay
we can do one more

41:47  if 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

42:03  is 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

42:32  cos 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

43:00  okay 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

43:33  cos 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

44:00  plus 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

44:36  alright i think everyone had enough for one day
