were gonna do more derivativing today
as just a reminder of what we did the other day
we learned the power rule
power rule is what you use with things like polynomials
y equals something to a power
with a constant in front is okay
then the derivative

0:30

you just bring the power down and reduce the power by 1
that was very simple right, the derivative of x to the 10
if y is x to the 10
then the derivative
is just 10 times x to the 9
if youre going to do the second derivative
which youll learn this notation later
is 90

1:00

x to the 8, and you can keep going
till you run out of derivatives, youll run out of derivatives at 10
so we can ask for the 3rd the 4th the 5th
the whatever derivative okay
and each time
you would bring the power in front so
jus to heat this up, you have y=x to the 4
1st derivative is 4x to the 3
second derivative is 4 times

1:30

3x to the 2
3rd derivative
is 4 times 3 times 2
does that look familiar
you know what function that is, the fourth derivative
4 times 3 times 2 times 1
x to the 0, what is 4 times 3 times 2 times 1
4 factorial
x to the 0 is just 1
the derivative of x to the 4th the 4th derivative of x to the 4th

2:01

is 4 factorial what do you think will happen with the 5th derivative
youll get 5 factorial
oh im sorry let me rephrase that
the 5th derivative of this will be 0, because the 4th factorial will be constant
so once your derivative passes that number
you just get 0
again if y=x to the 5th
then the 5th derivative
will be 5 factorial

2:31

if y=x to the n
then te nth derivative
is n factorial for those of you who dont know what n factorial is
is n times n minus 1
times n minus 2
all the way down to 1
so if you have n to the 100
100 times 99 times 98
97, all the way down to 1
okay very big number
the next derivative after that is 0
so if i said
whats the 100th derivative

3:02

x to the 99
its just 0
so far so good, thats the power rule
you can have some fun with that look for patterns
we normally ask for the first and second derivative, you can do lots with the first and second derivatives
we dont really care beyond that
unless we feel like just punishing you
NHE 127
youll learn about something called paler series
and thas where we use more derivatives

3:34

ah this is a notation thing so
theres another way to write derivative notation
dy/dx its called live denotation
what you are doing is doing d of dx
of y, which we shortened to dy
dx
so if you take the derivative of that
its d/dx
of dy

4:00

dx
which is d squared of y
dx squared thats why you use that notation
the third derivative would be d3y and dx 3 and so on
but you should get very comfortable
live notation because youre going to use that
much more than the prime notation
thats just a quick short hand but a lot of things
you really have to put it in this form
to work with it

4:32

we all understand this but more important
y=kx to the n
y prime is knx to the n-1
then we did product rule
product rule
when you have two functions multiplied together
when you have y is function f of x

5:02

times function g of x
and the derivative
is the first function
times the derivative of the second function
plus a second function
times the derivative of the first function
or the other way around
becaus your doing multiplication and addition order doesnt matter
so you can switch these two thinga

5:30

for example
if you had f of g
t cubed e to the t, oh wait getting ahead of ourselves
t cubed times 5t times t squared
now you can certainly just distribute that
but if you wanted to do the prouct ruel

6:03

f prime of g is the first function
times the derivative of the second
which is 5 pkus 2t
plus reverse
derivative of t cubed is 3t squared
times 5t plus t squared

6:37

then we learned derivative of e to the x
which is e to the x
that is very challenging but many of you will see to memorize that
the derivative of e to the x is e to the x
the second derivative is e to the x the third derivative is e to the x

7:03

its just e to the x
isnt that handy? very good to know
e to the x also, e to the x is never negative or 0
because you remember what the graph of e to the x looks like
the graph of e to the x looks something like that
so notice its never 0 and its never negative
this will be very useful because remeber when i told you youll have e to the x

7:30

youll have an equation and find out where its 0
if e to the x is the equation, e to the x s never 0 so you can just take it out
so where would that show up
suppose you had f of x
x cubed
e to the x and you wanted to take the derivative
notice its two functions so you would have to use the product ruel

8:00

this is something people trip up on all the time
so the derivative is x cubed times the derivative of e to the x
plus e to the x
times 3x squared
notice e to the x is in both of those terms
so you can pull that out

8:33

in fact you can also take out an x squared if you wanted
if you were setting this equal to 0
notice x squared can equal 0 at 0
and x+3 can equal 0 at -3 but e to the x is never going to be 0
cause e tot he x is always positive
so if we had to find the 0s

9:00

you would just find where this is 0
this is 0 and ignore the e to the x
we go that
dont go it, what dont you get
so far so good
another word we learned the quotation rule

9:41

is you have f of x over g of x
this one is lidehi-hidelo
the derivative
the low function
times the derivative of the higher function
minus the high function
times the derivative of the lower function

10:00

over the lower function squared
so for example

10:33

you want to find the derivative of that
you take the derivative
you use the quotient rule
bottom function
times the derivative of the top function which is
3x squared
3x squared plus 4 minus
top function times the derivative of the bottom function

11:01

so x cubed plus 4x
times 2x
all over
x squared minus 1
squared
thats not so bad
the messy part is when you have to do a second derivative
because you have to simplify theses things, lots of times it simplifies quite nicely

11:32

lets practice this one you would multiply out, youd get 3x to the 4
plus 4x suqred minus 3x squared
minus 4
minus
2x to the 4, minus 8x squared
all over
x squared minus 1
squared by the way when is a fraction equal to 0
A fraction is equal to zero When the numerator is equal to the

12:01

Provided the denominator is also not zero
When you have to find zeros you only care when the top is zero
What makes a fractions zero the bottom is another thing
So you'd simplify that let's see
You get x to then4
X squared-7 X squared, 7X-4 to the fourth
Which is easier to work with over

I am going to wait a few more seconds and then I'm going to the erase this

13:32

Okay we have to remember our trigonometry for a couple minutes
Remember the unit circle
Something like that I'm only going to draw a quarter of the circle
Just the first quadrant you don't need the whole thing

14:03

There you go that came out better okay
Let's call that o for the origin
And remember when you do circles you have a radius that sticks outside from the origin
I'm going to label some of these points call this point B
And we're going to call this a where intercepts at the X axis

14:30

It's a unit circle unit meaning one the radius is one
Which means this distances also 1
OP is one an OA is one
How do we find the coordinates of point P
We have the x-Cordinator in the y-coordinate so if you remember from soa-cah-toa

15:05

This distance is cosine theta because it's adjacent over one
And this distance is the sign of theta
Where is the tangent of theta
Turns out what letter did I call that t
This is the tangent of theta

15:32

thats why we call that tangent if you ever wondered why
thats the tangent of a circle
now you know, this makes a right angle
and why is that the tangent
well you think of this angle, this is theta
tan theta is opposite over adjacent

16:04

which is ta/oa
but oa is 1
so the tangent of theta
is ta/1 so this length
is the tangent of theta, you can think of the unit circle stuff
thats the sin thats the cosin thats the tangent
and if you think of this arc
as that arc gets closer and closer to 90 degrees
that line is gonna get higher and higher part of why the tangent
goes up to infinity

16:32

in case you ever wondered
one last thing lets draw
that line ap
pa depending on where you grew up
alright sp notice a couple things
the area of triangle
opa

17:01

has to be less than the area of this whole sectore
the sector opa
which has to be less than the area of the triangle
all agree so the triangle
is less than the whole sector cause you have this shaded piece out here
and the sector is less than the big triangle

17:36

well what is the area of the triangle opa well its 1/2 base time height
1/2. whats the base, the base is oa which is 1
and the height is sine theta
see that in color

18:04

this triangle
has a height of sine theta and a base of oa
so far so good
the area of the big triangle well if the base is 1
and its height is tan theta

18:32

how do we find the area of a sector, you guys remember how to find the area of a sector
lets review that
you have a circle, a pizza you want ot find the area of a slice of pizza
its just proportions, if you know this radius

19:02

and you want to find the area of that slice
the ratio of the angle
to 360 but were doing this in radians so 2pi radian
is the area of the piecd
divided by the whole circle so again
the angle compared to the whole thing
equals the sector compared to the whole thing
cancel pis

19:34

and you get 1/2 theta r squared
equals the area
or r squared theta, the area of that circle is 1/2
theta r squared or r squared theta
theta is in redians and by the way in calculus
everything is in radians
were not going to do anything in degrees sorry folks
thats why you learned radians last semester in trig

20:02

1/2sin theta
is less than 1/2 r squared theta
is less than 1/2 tan theta
so far so good
so we can cancel the 1/2
and you get sin theta
is less than r squared theta
is less than tan theta

20:39

lets divided everything by sin theta
so this is 1
this is r squared oh im sorry
well do it next set, r squared theta

21:00

over sine theta
and what os tan theta. sin theta that simplifies
sin/cos/sin lets see
sin
over cos
divided by sin
is sin
over cos
times 1/sin
is 1/cos

21:30

okay so thats 1/cos
by the way what is the radius of the circle
you said this is the unit circle so the radius is 1
so we can take r squared and replace it with 1
were starting to get things good here
1 is less than theta over sin theta
is less than 1/cos theta
okay flip everything upside down
and you get

22:03

1 is greater than sin theta over theta
is greater than cos theta
thats called the reciprocal okay
thats a fun thing, alright what happens when theta goes to 0
is you want to do the limit now as theta goes to 0

22:30

1 is greater than this thing
whats the cos of 0, 1
thats the squeeze theorem so that tells you
that the limit, this is the goal
the limit when theta goes to 0
sin theta over theta
equals 1, providing we are doing things in radians
so all of that was to make sure you know
and understand this, thats enough theory i like to do

23:02

but i like to throw it in once in a while to keep everyone honest
r squared 1 for 1 cause the radius is 1
so once again thats a proof
the idea ia because were going to need this in a couple minutes
to make sure that you understand that the limit
as theta goes to 0, sin theta over theta is 1
we like to test this

23:31

so lets put that some place safe
im going to swith it to x cause it doesnt rally matter
very important limit to know
make sure you do because we will test you
in some variation of this
a couple of other things that you might not know
suppose we want to ind

24:02

the limit as x goes to 0 of 1-cisx/x
what would you do
well take some guesses
throw out some numbers pi, 1
0, infinity, cant be determined why dont i wait for professor kahn
you think its 1, who agrees its 1, who thinks its 0
why dont we multiply the top and the bottom of this

24:34

by 1+cosx
what do you get then, okay so now
the limit as x goes to 0
1-cos times 1+cos is the conjugate so its cos squared
somewhere in the back of your head that should sound familiar

25:00

1-cos squared is sin squared
oh what do we do with that, we can break that into 2 limits
we can break that into
limit x goes to 0
sin x over x
times sinx

25:32

over 1+ cosinx
the limit as x goes to 0, of sinx/x we just learned that equals to 1
so this is the limit as x goes to 0 is 1
whats sin of 0
0 and whats 1+cosx
2 because cos of 0
is 1 so that times 0
is equal to 0
anybody who agreed with me

26:02

got a 0, good
thats our second q
you guys are smart to wait
were building blocks for learning stuff
im going to now erase this

26:31

last thing, angle addition formula
remember this from the regents that was on the sheet
that you can break and use through the exam
so you didnt really have to remember it
sina cosb
plus cosasinb
remember that
regents, algebra trig regents

27:00

howd we feel about that one
we liked that one

27:35

why am i doing all of this
because were gonna learn now what the sin of x is
lets find the derivative of sin of x

28:00

were gonna learn what the derivative of sinx is
then you can tear out the other pages in your book and say im just gonna memorize this
derivative of sinx, lets do the limit
as h goes to 0
sin of x plus h
minus sin x
all over h

28:39

we gonna do this as a team
okay sin of x plus h
thats this thing, thats gonna be sinx
cosh

29:00

plus cosx
sinh
minus sinx
all over h
cause you use this formula, this formula, im not gonna tell you where this comes from
its an mat 123 thing not a mat 125 thing
im just going to do a tiny bit of rearranging, im gonna more the 2 and the first terms

29:35

and rewrite this now as limit as h goes too 0
of sinx
sinx, cosh
minus sinx
plus cosx sinh
over h, i didnt do anything i just flipped the second and third terms
now lets break that into 2 limits

30:06

first two terms i do sinx
cosh minus sinx
over h plus the limit
h goes to 0 cos x
sin h
over h
okay so the left one and the right one

30:34

now i can factor a sinx out of that
sinxtimecosh-1/h plus
limit as h goes to 0
cosx times sinh

31:00

over h
well the x terms dont contain h's
they just stay the way they are, when x goes to 0
the limit of cosh-1/h
is this limit
which is 0
this part becomes sinx
times 0
and here again cosx nothing happens when it goes to 0

31:30

the limit as h goes to 0, sinh/h
is 1
what have we learned
we learned that the derivative of sinx
is cos
so if y is sinx
dy/dx
cosx, you guys can remember that without to much work either

32:10

derivative of sinx is cosx
almost so
now i wont prove this one but the derivative of cosx
is the negative of the sin of x

32:49

this top graph
is the graph of sinx which you all memorized
long time ago

33:00

lets think about what that looks like well whats the slope here
remember now x goes to 0, sinx over x
is 1
x and sinx are kind of doing the same thing
and therefore the slope here is about 1
and now the slope is 0 about hre

33:30

so we have positive values
until we get to 0
now lets look at the slope, theyre negative numbers
until you get to here where its back to 0
and then theyre positive numbers again
and then repeat thats the cos graph
so the derivative of sin is the cos
isnt that awesome, you guys are like math people thats really goof

34:10

these are important to memorize
alright how bout some more fun ones
lets do the derivative of tanx

34:35

how can we do the derivative of tanx, im not gonna do it the hard way
ti much work, besides i did all that work to do the derivative of sin
well tanx
is sinx over cosx
so if i wanted to do the derivative
i use the quotient rule

35:02

lidehi/hidelo over lo squared
derivative of the bottom cosx
times the derivative of the top
well we just learned whats the derivative of sin, cos
minus the top
sinx
times the derivative of cos which is -sinx

factor out the minus 1
and you get minus
sin squared plus cos sqaured
over sin squared
which is negative 1 over sin squared
which is negative
negative cosecant squared

38:53

two left
suppose we have y=cscx

39:01

well that 1 over cos
this is minus sin squared minus cos squared
take out a negative you get sin squared/ cos squared
thats negative 1 over sin squared
negative csc squared
how do we do the derivative of tht one
well we havent learned other stuff yet so lets do the quotient rule

39:40

lo derivative of 1 is just 0
minus hi time the derivative of the bottom
over the bottom squared

40:09

cos x times 0 is 0
so this is just sinx
over cos squared x
but we can make that 1/cos
times sin over cos we can think of it that way

40:32

1/c=csc
sin/cos=tan
we get y=secx
dy/dx
is secx
tanx
how are we going to remember that

41:19

1 more
y=cscx
so we can make this one 1/sinx

41:30

and then you get dy/dx
equals sinx
derivative of 1 which is 0
minus the derivative of sin 1 times cos
over sin squared
or negative cosx
over sin squared x

42:02

you can now break that into negative 1/sinx
times cosx/sinx
or cscx
cotx
so last one
y=cscx
dy/dx is negative csc

42:31

cot
so notice some patterns
you take the derivative of the trig function
if you take the derivative of its co function
you get negtaive with the co functions
when you have the derviative of sin the co sin co function
the derivative is -sin which is the derivative of cos
when you have tan you get csc
when you get tan you get -csc sqaured

43:02

you get sec you get secx tanx
when you get csc you get negative
cscx cotx
memorize these ill put some pages from my book up on blackboard
soon alright lets do a little bit of practice
nothing to hard, we save the hard stuff for the exams

43:42

whats the derivative of that
lets do the derivative of e x sinx

44:02

well lets see the product rule first function
times the derivative of the second function, whats the derivative of sin
cos
the derivative of e to the x is e to the x
sinx
that one wasnt so bad was ir

44:44

how bout that one
what is the tanx over suqare root of x
alright you ready, the derivative

45:00

low
dehi, derivative of tan
csc
squared minus tangent
times the derivative of the square root of x, did you memorize the short cut to that
i hope you did, 1/2x
over the bottom squared which is x
lets add that to the pill of things we have to remember

45:37

i wouldnt let you take in a card with all this but feel free to tattoo
the other one you have to memorize is this y=1/x
dy/dx
is negative 1/x squared

46:26

this is a fun one, lets take the derivative of this

46:58

this isnt very hard this is just regular stuff

47:02

so dy/dx
is 20x cubed
minus 6x squared
the derivative if cot is csc squared
so minus 2csc squared
6x thank you
the derivative of cot is csc squared

47:30

the derivative of -cscx is the derivative of csccot
so plus 4
csc cot
remember when you take the derivative of csc and sac
it returns to the derivative
wait till we have to go backwards and do it the other way much harder
that wont happen till 126, so that wont happen for a while
alright i think thats enough for 1 day