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welcome back as this is the first of several presentations we were going to
introduce some further differentiation rules that go beyond the poem roman
differentiation of polynomials
and i wish we will then put to use to show how and when and how different
cation who works
and how on the other hand weekend reinforced
the fear is ideas uh... that can be computed with calculus
so let's take a look at the product and cool people
uh... we need the product in the courtroom rule basically because there
are certain products like you to the extent that script as to which cannot be
multiplied out like
products that you may have seen uh... as you were working with the power rules
such as
uh... execute times expect us to with the idea is well first multiplet
parentheses
and then apply the powerful
uh... civility cautions a highlight rational functions so to execute
xc request three over
expert months for that cannot be converted
into functions that don't involve portions and so
we need to have rules that allow us to just difference here
abstract products in court resolved functions
uh... what should they look like
well it's it's very tempting after the summer rule to hope that you can just
move the terrific inside
n dot this line is just designed to make sure
don't things that
because we must be really careful here of the greatest can't be moved into
practical shins
if we take for example g affects people's excuse me
and chaotic sequels expect us to
well then the product of g and katie's executed in sixth grade class too
if you're not cut out to get it to the fifth class two x
and that you refer to both sides is of course five x two the force
uh... classic six square
if that's not equal to use the product off the derivatives because product of
the president's would be three x quitter
times two x
and uh... wells at six x cuban that's certainly not
expression up here
walking
so
we
me to go back and think about
how to derive the product rule again mrs
the idea behind proofs essentially even tho mathematics is often present as he
was a fear of usa proof
the proof typically exists before the theory more sometimes definitely exists
before theory because
the proof in some way leads up to this year and that's the case for the product
will sell
the only way we know how to really computerize it hits even though it's
it's not terribly pleasant necessarily
is a different school content so
just like any derivatives the derivative of the products of g_-ten skate prime
is just a minute if they chose to zero g_ can fix because it's not his cheek
text-align wage
so how do you work that out well first of all
you resolve the product right cheek affects plus a chance
gaf exposes
ko explicit
aunty ko fixes to takes care of x
and know what we're looking for a i would be different portions that used to
fix plus six months g_o_p_ statement by h okay if its message plants catholics
decline by h
and um... well we can't work that out with this expression because week in
fact around the common terms
if we had an extra term say
uh... geo x timescale next class h that with the fact racking up explicit agent
that's exactly the idea
for the proof
the only thing that happens in this step even though it may look at it
intimidating is
uh... remember there are two tricks in mathematics padding zero px multiplied
by one
but we've got zero here right these terms just add up to zero is just said
these terms happen to work out in such a way
that's
we can talk or things out much fitness is again something
that you would be
training in detail in a proof last year which is doing this for background but
what we can say yes
that
universe terms tony
captured all of you
in the first terms
weekend factory ko expose
into in c
careers at the end
we can factor out ngo nixon that's nice because that leads us to use that as it
goes to zero g_m_ picked us eight months to your pics quantity time scale exposes
ko express h minus ko fixed
times geodesic
quantity times geo testified by a_g_ here we can see that we've got different
quarter four g because these things the right by h
and we've got a different political katie christensen
is defiantly so we can puzzle the fraction support
we get the benefits goes to zero out the difference quotient fujita scale
explicit h
plus the difference quotient for katie times g of x
and uh... we can pull the limits insight now cause and it also allows that
and i whether we can see the limit
of the difference quickly he gets his jeep wrangler fix that in itself care
experts say it will be camp x
that ended up the difference quotient of katie
gives escape crime affects epidemic geo fixture geo trax
derivative off the product
g k
ends up being g primetime skate escape route
times g
and i think a revision of the chronicle off the course in coral
is indeed quite similar so we'll
forego that part remember these
derivations are for background information we want to nowhere that
stuff comes from we want to realize that no this is not something that somebody
opportunity right
wrote down this can be justified
but even begins
the idea of a calculus classes to apply the swirls network id
so first yes just uh... summarize these rules theory is
uh... about the product and a quarter rules
in if you've got a product equals determines chaotic warship you equals g
overtake
where do you care differentiable at the point ics
well then
the derivative of the product
is g crime
tim's cane class k primetime gene
and if catholics is not able to see a resident of direct derivative off the
court reviews
g primetime scary
minus k primetime g divided by case where
and the way to remember these is you want to make
sure that you remember the product rule in the same water and stir numerator off
the quotient wrote
basically because that way
you don't have to worry too much about
these negative signing here
i have seen get also an armenian exam
in and typically it was because
student remembered the product rule one way and then try to remember quadrupling
of the women things just japanese fund
fact when i was a student the only thing i memorize was the quotient rule
and i realized and i remember the product was just enumerated with glass
so there are many ways to
tell your memory out here and the main thing is he does
need to have these rules available
is you were doing mathematics
you can also
write these rules are thoroughly and so for example the product rule simply says
that
the derivative of a product this incredible first factor
times the second factor plus security for the second factor
times the first factor
and all of that yeltsin because
if you try to just start
naming a bunch of stuff
well that's what we're running into this thing that i've talked about in the
introduction to the course ukraine has about seven spaces for memory and if
you're using a two of those for new letters
well then you could only find spaces left to do the rest of the problem with
and that he had
after things up a bit and and so we can be
very uh... beneficial to not think of the product as a g tim's cane but as a
first track record setting factor
civility
it can be beneficial to think of the quotient not s_g_ over cable s
numerator over denominator
and then the quotient rule says
that the derivative of quotient is to get rid of off the numerator
tyson nominated
minus it riveted off the numerator director reveals economic i'm sorry
times the numerator
and then we divide by the square off this moment
so for quite a few ways to you
keep these rules in mind the main thing is to find one
that works for you because you are about to compute the strictest rightly
now that we have been well let's just computes on derivatives dot
simplest kind of problem in this setting that was just say take a function
computed derivative supports out here because
each of the exxon six square close to him
invigorated off that would be such a great deal of each of the x
times expect us to
placid a review of expect us to
times each of the x
and that's of course just need to the extent expect us to cuts two x
times each of the x
and the only thing we can do you know we have simplifications factor of the to
see you next week it
it to the extent expect us to explode stew
uh... aitken if you see a factorization attributed
you might as well go ahead
and executed because from what we've already seen about calculus
under a lot of times were interested in two zeros off the directly to consumers
are easier to find
for a fact expression
but that does not mean we're just starting to
run around and try to factor
anything just come what may if we see one
we put it down if not we just need the expression issues in singapore
okay what else can we do what we can commit to reduce of quotient spreading
much the same story only that we have to
used to quote unquote
so if primer thinks he would be such a record of off the numerator
time sitting on a mentor minus attributable to denominator
tyson numerator
divided by the square off the denominator
typically we don't
writes this stuff down even though at the beginning i'd like to
all the steps possible
typically the first step is to albany
computer derivatives inter
numerator and so you end up with security be forex times
excoriate minus for months two weeks
times to its crew plus three different banks glickman's work one of the squared
and then really the only thing we can do here is
multiply things out i i didn't even factor of the x_ here i don't think
yeah this is just four x q minus sixteen x
minus four excuse
and i'm i'm a six six and you can see for its cuba switch goes away
litany of twenty two x normally services
made his way to explain it but it's great minds for
quantity squared
and i'd look at the other
good business advice here is that when you differentiate quotient
eat is advisable to leave the denominator as our because typically
we do
uh... especially in drafting we will be interesting also competing sake
interactive
and it will be advantageous
to have this as a power because then
atlanta chain rule
and we will see a few cancellation are helpful
uh... more easy that way so these economic times out
blessed me with darkness so we could compute
equations often two times for example for function like extends each of the x_
which looks rather harmless but remember we really need the chronicle
to differentiate this thing okay so we want to find the questions attention
about this function
an equal split what do we need we need to prove it
uh... interactive is diverted over x which is one thirty two the x
class x considerate of each of the x_ which is eating itself and so that's
just
into the extent six plus one
we need to drift at the point of the word to complete the tangent line
so that ex-prime of one is just
tui each of the first time twelve plus one
we need to cut you off the function of the point which should be just be
radical one this one tells you to come on
so that he
and uh... wells a week and we cast
the equation of the tangent line either with until formulas that you've
memorized
as i would always recommend justice
to memorize too much fun necessary stuff instead
get used to executing routine algebra it's just math
the equation for straight lines wapas aane explicitly that in fact has
so m equals to me it goes to the point one
which means we get that you speak with a two e times one
times beach
solvent for buettner with bb native in
and that means that our
equation of the teens clients why equals to be x
minus p
looks a little bit more complicated and other teams brunswick worked out before
but remembered use just
uh... oilers number two point seven one eight to eight one eight two eight four
five nine zero
for friday and so on and so wanted somebody zita just with eke
uh... we can also double check this problem by
graphing the function in paragraph in the proposed tangent line but here's the
function
here's the point where we're going to compute the team declined and here's
lines at the computer than the previous panel and we can see
that yes this line ought to be correct because it but it does look
lack attention plan should look
bucket we can also by using calculus two ten where function is increasing or
decreasing and with this being a rational functional quotient rule
so that riveted yes
well there are a little bit of both the numerator one tujhse denominator export
plus for the quantity
minus derivative profited nominated or two x
tyson numerator x divide it by
denominator x quick plus for quantity
okay and that we just simplify replicate x squared lines to equip it should be
minus x squared so
for money six rhetoric squared plus four-party square
and we can factor as active quest for my sixth graders to plastics to monday six
to my sex to classics
the denominator we can factor because in fact the denominator does not having
zeros
so that already means that the degree that it has a cigarette to erect base
here at minus two
it's defined everywhere so we only have two critical points plus minus two lb
sits at the sequence zero
two critical points plus minus two
we do our number line
two critical points native to entrust to
we need three checkpoints at the quest to land mines two are so far apart from
each other
we get nice checkpoints which negative three at zero
and at three
and we need to find the signs of the derivative
well let's see if the derivative
at needed three what can we do here
uh... the derivatives the denominator here
denominator is always positive or non-negative infact it's always positive
so we don't have to worry about three billion only have to worry about the
sign up for my six quit
well for minors mega pics trees x
for minus negative three squared
is are four months nine which the faisal it's negative there
four months zeros for which is positive
in four minus ninety is negative again
so we get that to function should be
decreasing increasing in decreasing
and again
we can double check that by simply graphing the function
and comparing the graph to our
analytical result in base when i think a crew rather well to scale because
well negative to was here plus who was here and we can see you believe the
functions decreasing
increasing in decreasing think that
that took me quite a while to just make the adjustment recompile make the
adjustment we compiled all that
uh... baggage
it just helps
to verify things it's a good sanity check as always
almost literally creech don't worry about the back of the book
find ways to check your work
that you can also executes when you are on the job war
when you were taking an exam eve
graphing calculators in this case are allowed on the exit
okay finally we're finishing up with any
more complicated derivatives that goes in the direction of
giving us accustomed to the fact that sometimes
these differentiation rules pile up in this is another thing that
sometimes see being a bit of a cliff on a test of that yes
in expression such as ever thanks equals excluded each of the x
times extras when people realize this is an equal shinde
inside the quotient the numerator isn't the product which means
the jury beautiful things numerator
must be computed with the product rule ain't no other way
and all that is where sometimes people just obj and a little bit too quickly
and just take the product of the tradition of course
that's wrong
so the derivative here is computed with the court has ruled
within the courtroom rule
these
derivative of the numerator we can see is computed
with the product will apply to every beautiful xd greatly to the exceeds
to exeter and above the first doctor times each of the x_ which is the second
factor
plus each of the expert first factor times
after derivative of the first
second parker
tines
ache squared which is the first time list is a
the derivative off that first factories two x
the same track risky to the extra credit about the second factories each of the
x_ also and the first factories exports of exhilarated both in the mirror
then we get
ties to denominator extra swan minus derivative of the denominator which has
fallen times in the water which is excluded to the eggs divided by the
square incident on air
in then it's just a matter of multiplying things out and factory
melded each of the excess inka konni first multiply help ramsey's know at
that they did both ok
we have a need to be exceed common here that comes out
then we multiply with the parentheses to extends x's two x where
x where times exit excuse
two x times one is two x exquisite times one music squared so i just nag activity
don't remember foil first however in our hostages remembered that
when you multiply prevent disease everything in the first parenthesis has
to be multiplied everything the same apprentice
and that means with
differences of two times and the president to you get four factors
uh... for summons and that's what got him
and uh... well the one of course you superfluous we just get my six thirty
two b x and we've got an ex post once quality screen in ahmedabad change
and so the final result here is a company to the exceptional lighthouse
we get three x squared minus six credits to exporter of plastic skew class
two x so it should be
but we have here because the aches
times execute
plus two x squares
ethics guidelines expert was away
class two x divided by experts warn quantities cleared
that's about as good as it gets that derivative is done
okay well this is it
for their product and quotient roles we've gone through
pretty much all the applications that we have four there it was a conceptual
things except for computing call cavity
except for graphing and except for optimizing she graphing uh... off
rational functions specifically will talk about in the next presentation on
the chain of rule
the chain rule allows us to up compute the second derivative of rational
function of the easier
and i think a lot of time optimization we talk about the channel
but for this one
go ahead and just practice practice quotient rule
intend move along