WEBVTT
Kind: captions
Language: en
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we had the chain rule
00:00:05.960 --> 00:00:08.520
the chain rule drives some of you crazy
00:00:08.520 --> 00:00:10.680
so before we do differentiation
00:00:10.680 --> 00:00:12.780
were just going to practice the chain rule for a few minutes
00:00:13.420 --> 00:00:18.060
so remember to do the derivative of the chain rule you have 2 functions
00:00:24.620 --> 00:00:27.580
f of g of x you could actually have 3 functions
00:00:28.180 --> 00:00:31.300
r 20, you can mess this or clean this how you want
00:00:31.300 --> 00:00:33.940
but the principle all you need is 2 functions
00:00:34.360 --> 00:00:35.640
you have an outside
00:00:35.780 --> 00:00:36.980
you have an inside
00:00:37.760 --> 00:00:41.520
in order to do the derivative you do the outside function
00:00:41.620 --> 00:00:43.380
you leave the inside alone
00:00:43.640 --> 00:00:46.600
times the derivative of the inside function
00:00:46.940 --> 00:00:48.060
so another words
00:00:48.920 --> 00:00:51.880
you take the derivative of the outer funtion
00:00:52.340 --> 00:00:55.140
you dont do anything to the inner function
00:00:56.340 --> 00:00:59.300
times the derivative of the inside function
00:00:59.300 --> 00:01:03.260
we i know a lot of you, the next webassign will have a lot of chain rule function
00:01:03.480 --> 00:01:06.440
not due for a couple weeks so better work on it
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dont do it till you feel comfortable
00:01:10.600 --> 00:01:13.400
so heres the type that trips people up a lot
00:01:19.520 --> 00:01:20.880
something like that
00:01:21.620 --> 00:01:24.660
so take a second and see if you can figure it out
00:01:25.640 --> 00:01:27.480
this is puzzling lots of you
00:01:29.940 --> 00:01:31.460
theres 3 layers to this
00:01:31.660 --> 00:01:33.500
because if you rewrote this
00:01:34.080 --> 00:01:35.680
this is really cos of 10x
00:01:37.920 --> 00:01:38.420
cubed
00:01:39.140 --> 00:01:42.660
so you have 3 things going on, so always ask youreself
00:01:42.700 --> 00:01:44.140
what is happening to x
00:01:44.420 --> 00:01:46.660
first i take x and multiply it by 10
00:01:48.460 --> 00:01:50.620
second i take 10x and take the cos
00:01:50.940 --> 00:01:53.900
and then third i take cos of the 10x and cube it
00:01:55.060 --> 00:01:57.060
you got three layers, f g and a h
00:01:57.220 --> 00:01:59.380
so how would we do the derivative
00:02:01.020 --> 00:02:02.060
1 layer at a time
00:02:02.060 --> 00:02:06.000
but first im just doing the derivative of the something cubed
00:02:06.720 --> 00:02:08.960
so its 3 and the something squared
00:02:10.280 --> 00:02:12.680
now im going to fill in the something
00:02:13.140 --> 00:02:15.700
that doesnt change, thats just cos10x
00:02:21.040 --> 00:02:23.040
then i do the derivative of cos
00:02:23.920 --> 00:02:25.360
which is minus sin of x
00:02:26.920 --> 00:02:28.680
notice i still have the 10x
00:02:29.220 --> 00:02:30.740
but that hasnt changed
00:02:30.740 --> 00:02:35.380
notice their isnt any squaring anymore its just the derivative of cos
00:02:36.200 --> 00:02:38.840
hen i do the derivative of 10x which is 10
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much easier to watch me do it
00:02:43.440 --> 00:02:45.040
right good so got the ide
00:02:45.140 --> 00:02:46.420
lets do another one
00:02:53.100 --> 00:02:55.740
almost the same but slightly different
00:02:57.740 --> 00:03:01.260
this time we have 3 layers again just like the last one
00:03:06.020 --> 00:03:07.860
we have something to the 4th
00:03:08.980 --> 00:03:10.820
we have sec and pix so you say
00:03:10.820 --> 00:03:13.440
what am i doing to x, first i multiply it by pi
00:03:14.360 --> 00:03:16.680
then i take pix and take the sec of it
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then i take the sec of pix and raise it to the 4th
00:03:19.780 --> 00:03:21.620
3 levels just like last time
00:03:23.280 --> 00:03:28.160
so the derivative, first i have to do the derivative of whatever to the 4th
00:03:29.660 --> 00:03:31.020
this thing to the 3rd
00:03:35.560 --> 00:03:36.200
its just 4
00:03:37.140 --> 00:03:38.020
sec pi x cubed
00:03:38.080 --> 00:03:40.320
now take the derivative of sec pi x
00:03:41.140 --> 00:03:43.860
this is where people mess up thats sec pix
00:03:44.880 --> 00:03:45.680
tangent pix
00:03:51.280 --> 00:03:54.560
then you have to do the derivative of pi x which is pi
00:03:58.000 --> 00:04:00.560
who got that, that makes me feel better
00:04:01.320 --> 00:04:02.440
lets try one more
00:04:19.860 --> 00:04:20.420
last one
00:04:23.720 --> 00:04:27.000
so this is another 3 layer one again, but this time
00:04:28.120 --> 00:04:29.400
i use the cubed root
00:04:31.380 --> 00:04:33.300
this is cot of x squared plus 1
00:04:35.340 --> 00:04:38.540
to the 1/3, because cubed root is to the 1/3 power
00:04:41.020 --> 00:04:45.020
so first i have to take the derivative of something to the 1/3
00:04:47.800 --> 00:04:48.600
which is 1/3
00:04:50.620 --> 00:04:52.060
cot of x squared plus 1
00:04:52.100 --> 00:04:54.500
so you dont do anything to the inside
00:04:56.800 --> 00:04:58.240
now 1/3 this minus 2/3
00:04:59.920 --> 00:05:01.680
because 1/3 minus 1 is -2/3
00:05:01.680 --> 00:05:05.940
this is where you guys will trip up, youll do 1/3 - 1 and youll do that wrong
00:05:06.000 --> 00:05:09.280
okay so practice make sure you get the hang of that
00:05:10.120 --> 00:05:12.440
then you have the derivative of cot
00:05:14.020 --> 00:05:16.020
which is negative cos squared
00:05:16.020 --> 00:05:18.280
and you dont do anything to the inside
00:05:20.940 --> 00:05:24.060
and then times the derivative o x squared plus 1
00:05:25.360 --> 00:05:26.080
which is 2x
00:05:27.400 --> 00:05:28.840
howd we do on this one?
00:05:30.520 --> 00:05:31.240
not as good
00:05:32.340 --> 00:05:33.380
not as good okay
00:05:33.380 --> 00:05:35.440
so we still have quite got it down yet
00:05:35.620 --> 00:05:37.140
but were getting there
00:05:38.520 --> 00:05:41.400
how much time do i have, i have 5 more minutes
00:05:52.320 --> 00:05:52.960
how about
00:06:01.720 --> 00:06:03.880
lets take the derivative of that
00:06:06.040 --> 00:06:07.640
this is going to require
00:06:08.260 --> 00:06:12.260
the product rule because we have 2 things going on, we have 5x
00:06:13.340 --> 00:06:15.740
say to yourself whats going on with x
00:06:16.820 --> 00:06:19.140
you take the cosx and do the e of that
00:06:19.240 --> 00:06:21.560
and thats all you can do tp that step
00:06:22.420 --> 00:06:23.940
this x you multiply by 5
00:06:23.940 --> 00:06:27.140
so these are really two different things that are happening to x
00:06:27.140 --> 00:06:28.220
so thats how you know
00:06:28.220 --> 00:06:30.660
its a product rule because its two different functions
00:06:30.880 --> 00:06:32.400
okay so ill repeat that
00:06:32.940 --> 00:06:36.140
x you take the cos of e and then take the e to the cos
00:06:36.140 --> 00:06:38.000
so thats 1 thing being done to x
00:06:38.000 --> 00:06:40.640
and the separate thing is multiplied by 5
00:06:40.640 --> 00:06:42.700
thats why you need the product rule
00:06:45.420 --> 00:06:49.980
you will almost always need a chain rule to do the derivative because
00:06:49.980 --> 00:06:51.920
the only time you wont need chain rule
00:06:52.080 --> 00:06:54.400
is when you just have a function of x
00:06:54.980 --> 00:06:56.340
so if you have sin of x
00:06:57.240 --> 00:06:58.520
or tan of x or log of x
00:06:58.520 --> 00:07:03.120
then you wont need the chain rule but as soon as theres a number in there, sin of 5x
00:07:04.440 --> 00:07:05.880
or the natural log of x
00:07:05.880 --> 00:07:07.700
squared plus 1 or anything like that
00:07:07.700 --> 00:07:08.900
now you can do the chain rule
00:07:09.780 --> 00:07:13.380
product rule first times the derivative of the second
00:07:13.380 --> 00:07:15.040
plus 2nd times the derivative of the first
00:07:15.040 --> 00:07:16.800
you have the first function
00:07:17.340 --> 00:07:20.060
now whats the derivative would be of cosx
00:07:20.060 --> 00:07:22.200
well we have an outside function
00:07:22.880 --> 00:07:25.920
and an inside function, the outside is e to the
00:07:25.920 --> 00:07:27.840
so youd just do e to the whatevers up there
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derivative of e to the x is e to the x
00:07:32.800 --> 00:07:33.300
times
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the derivative of cosx which is negative sinx
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plus reverse
00:07:44.260 --> 00:07:45.860
the derivative of 5x is 5
00:07:46.680 --> 00:07:47.480
e to the cosx
00:07:50.040 --> 00:07:52.040
if you wanted to simplify this
00:07:57.120 --> 00:07:58.320
you could pull out
00:07:58.420 --> 00:08:01.620
e to the cosx, by the way you could also pull out a 5
00:08:03.760 --> 00:08:05.920
but you could pull out an e to cosx
00:08:06.200 --> 00:08:08.040
when you have a product rule
00:08:08.040 --> 00:08:11.500
with an e function in it, with an exponential function in it
00:08:11.900 --> 00:08:14.700
you will always be able to factor the e term
00:08:14.700 --> 00:08:17.180
out of the power rule after you get the derivative
00:08:17.500 --> 00:08:19.020
that term will survive
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in all of the terms if we take it out
00:08:22.140 --> 00:08:24.060
1st derivative 2nd derivative 3rd derivative
00:08:24.060 --> 00:08:27.920
thats going to be important because when you set this equal to 0
00:08:27.920 --> 00:08:31.420
youll know youll have this term youre setting equal to 0
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and e to the anything is never 0
00:08:37.160 --> 00:08:40.040
so remmeber that youll always pull this out
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we have time for 1 more, sure
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heres a nice sophisticated
00:09:19.620 --> 00:09:21.220
this is chain rule again
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cosx+1
00:09:25.920 --> 00:09:27.080
over sinx plus 1
00:09:28.840 --> 00:09:29.340
minus 1
00:09:31.500 --> 00:09:32.860
all raised to the 1/2
00:09:35.560 --> 00:09:37.800
gonna need some space for this one
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so you say to yourself whats happening to x
00:09:49.540 --> 00:09:52.020
well first i take the cos of it and add 1
00:09:52.020 --> 00:09:55.360
and in the second function i take the sin and subtract 1
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his is going to be a quotient rule
00:09:57.580 --> 00:10:00.060
then i take that whole thing and raise it to a half
00:10:03.220 --> 00:10:06.660
first i do the derivative if the outside which is 1/2
00:10:08.160 --> 00:10:10.000
whole mess and the minus 1/2
00:10:10.440 --> 00:10:12.360
and i dont touch whats inside
00:10:18.480 --> 00:10:21.120
now i have to take the derivative of that
00:10:21.360 --> 00:10:24.080
thats going to require the quotient rule
00:10:26.140 --> 00:10:28.140
lodhi, hidlo minus lo squared
00:10:30.080 --> 00:10:31.920
so its a low function sinx-1
00:10:31.920 --> 00:10:34.340
times the derivative of the high function
00:10:35.080 --> 00:10:36.360
which is minus sinx
00:10:39.320 --> 00:10:39.820
minus
00:10:40.480 --> 00:10:41.040
reverse
00:10:43.540 --> 00:10:46.020
leave the top function alone, cosx+1
00:10:46.020 --> 00:10:48.200
now we do the derivative of the bottom function
00:10:49.280 --> 00:10:50.160
which is cosx
00:10:52.580 --> 00:10:53.140
all over
00:10:54.480 --> 00:10:54.980
sinx-1
00:10:58.100 --> 00:10:58.660
squared
00:10:59.420 --> 00:11:01.740
you can simplify that if you wanted
00:11:02.220 --> 00:11:04.780
why would you want to siplify that well
00:11:04.780 --> 00:11:08.120
if i was really a horrible person id ask you to take the second derivative
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you really wouldnt want to do that
00:11:10.420 --> 00:11:12.420
you might need to solve it for 0
00:11:13.040 --> 00:11:17.520
otherwise you would just leave it alone, if this was a question like
00:11:19.440 --> 00:11:22.240
find the equation of a tangent line of pi/6
00:11:23.460 --> 00:11:25.620
you would think 1pi/6 right away
00:11:25.620 --> 00:11:28.180
cause you guys all have to the sin an cos of pi/6
00:11:30.920 --> 00:11:34.040
but if you dont have to do anything else you stop
00:11:34.200 --> 00:11:37.800
this however will multiple nicely, minus sin squared
00:11:37.980 --> 00:11:39.500
and a minus cos squared
00:11:40.220 --> 00:11:40.860
and have 1
00:11:43.140 --> 00:11:43.780
or minus 1
00:11:43.780 --> 00:11:46.060
so it would actually simplify a lot
00:11:46.060 --> 00:11:48.460
but you dont want to have to do that, yes
00:11:49.840 --> 00:11:52.240
what happened to a 1/2 well its a half
00:11:52.280 --> 00:11:54.280
this whole thing is a minus 1/2
00:11:54.860 --> 00:11:57.980
and you stop and now youre just doing the inside
00:11:58.420 --> 00:12:00.660
the otter funtion is the 1/2 power
00:12:01.540 --> 00:12:02.180
so its 1/2
00:12:02.600 --> 00:12:04.520
inside function to minus 1/2
00:12:04.520 --> 00:12:07.620
then you only have to take the derivative on whats the inside
00:12:09.980 --> 00:12:11.580
how are we doing on these
00:12:14.060 --> 00:12:17.020
you have a couple weeks, who go them all right
00:12:27.600 --> 00:12:33.280
now were going learn somehting called implicate depreciation kind of the chain rule
00:12:33.520 --> 00:12:36.320
theyre related its not kind of its related
00:12:44.980 --> 00:12:45.940
alright so far
00:12:46.040 --> 00:12:48.520
all the derivatives we have have been
00:12:49.160 --> 00:12:51.400
in the order of y equals soemthing
00:12:51.600 --> 00:12:52.880
you have a function
00:12:53.020 --> 00:12:54.460
and you got y isolated
00:12:54.460 --> 00:12:57.240
everything on the other side is a different period, x
00:12:57.340 --> 00:12:58.220
t or whatever
00:12:58.220 --> 00:13:00.860
what do you do when you have a function and you cannot
00:13:00.920 --> 00:13:02.520
seperate the variables
00:13:02.520 --> 00:13:04.200
so if you had something like
00:13:11.080 --> 00:13:15.720
and you have to take the derivative of that, and that looks pretty easy
00:13:15.960 --> 00:13:17.160
but the problem is
00:13:18.440 --> 00:13:19.640
is not, find dy/dx
00:13:20.740 --> 00:13:23.300
the problem is its not y= something of x
00:13:23.880 --> 00:13:24.680
and you cant
00:13:25.820 --> 00:13:27.020
easily solve for y
00:13:27.020 --> 00:13:31.020
probably if you do some really complex algebra you can isolate y
00:13:31.720 --> 00:13:34.680
but otherwise its hard to do so the technique
00:13:34.680 --> 00:13:36.460
call them implicate depreciation
00:13:37.100 --> 00:13:39.020
who teach you to define dy/dx
00:13:39.820 --> 00:13:41.580
or any derivative without
00:13:42.060 --> 00:13:42.860
isolating y
00:13:42.860 --> 00:13:44.760
then you get the answer whats called
00:13:44.760 --> 00:13:46.940
implicitly because it will be in terms of
00:13:47.740 --> 00:13:48.240
x and y
00:13:48.520 --> 00:13:52.520
its implicate because you dont actually know what x and y are
00:13:54.080 --> 00:13:55.120
so what do i mean
00:13:57.080 --> 00:13:59.640
so what does it mean when you have dy/dx
00:14:00.860 --> 00:14:02.540
dy/dx is really the slope
00:14:03.000 --> 00:14:04.840
its saying how does y change
00:14:04.840 --> 00:14:07.660
in respect x, another words how does y change
00:14:08.220 --> 00:14:09.020
and x change
00:14:09.940 --> 00:14:10.440
if i had
00:14:11.100 --> 00:14:12.700
if i had y= a function of t
00:14:13.880 --> 00:14:15.560
dy/dt would be how would y
00:14:16.440 --> 00:14:17.720
change as t changes
00:14:20.700 --> 00:14:21.260
if i had y
00:14:22.580 --> 00:14:23.700
as a function of w
00:14:25.200 --> 00:14:26.000
i would do dy
00:14:26.500 --> 00:14:29.860
dw, i would want to know how y changes when w changes
00:14:29.860 --> 00:14:33.700
but what if i had y=ft and wanted to find out the change in dx
00:14:35.160 --> 00:14:36.520
well now the problem
00:14:36.520 --> 00:14:38.520
happens over here, i dont know what to do
00:14:39.120 --> 00:14:41.680
because t has to be in terms of x somehow
00:14:41.680 --> 00:14:44.660
but that point implicate depreciation is going to be 4
00:14:44.880 --> 00:14:46.160
so heres what you do
00:14:47.400 --> 00:14:48.120
every term
00:14:48.120 --> 00:14:50.160
youre going to take the derivative of
00:14:50.820 --> 00:14:51.940
is multiplied by
00:14:52.420 --> 00:14:53.620
d of that variable
00:14:54.640 --> 00:14:55.600
dx in this case
00:14:55.620 --> 00:14:56.500
what do i mean
00:14:56.500 --> 00:14:58.880
i take the derivative of x squared which is 2x
00:15:01.340 --> 00:15:01.840
dx, dx
00:15:02.400 --> 00:15:04.800
because thats, how does that change
00:15:04.800 --> 00:15:07.120
as x changes becasue you could think of this
00:15:08.000 --> 00:15:10.000
of x,x i know that sounds weird
00:15:10.360 --> 00:15:11.800
that would have to be 1
00:15:12.580 --> 00:15:13.460
but ignore it
00:15:13.460 --> 00:15:15.940
then i take the derivative of 3y squared
00:15:17.740 --> 00:15:18.300
and i say
00:15:19.400 --> 00:15:21.320
how does y change as x xhanges
00:15:22.720 --> 00:15:24.800
then i take the derivative of 8x
00:15:25.940 --> 00:15:28.820
then i minus 8 how does x changes as x changes
00:15:29.460 --> 00:15:31.140
nothing special happens
00:15:31.600 --> 00:15:33.680
then the derivative of 4y cubed
00:15:34.400 --> 00:15:35.840
which si 12y squared
00:15:37.760 --> 00:15:38.260
dy/dx
00:15:39.420 --> 00:15:39.980
so again
00:15:41.300 --> 00:15:43.860
were trying to find how y changes when x
00:15:45.120 --> 00:15:45.920
changes so y
00:15:45.920 --> 00:15:47.000
is a function of x
00:15:47.000 --> 00:15:49.700
so i do the chain rule, so the otter function
00:15:50.660 --> 00:15:51.380
would be 6y
00:15:51.380 --> 00:15:54.760
and then inner function would be how does y change with the effect of x
00:15:54.760 --> 00:15:56.160
which is right because you want it at the end
00:15:56.160 --> 00:15:58.480
when i do the derivative of x squared
00:15:58.780 --> 00:16:00.460
its the otter function 2x
00:16:00.460 --> 00:16:02.080
the inner function is just
00:16:02.080 --> 00:16:05.620
x so there is no inner function thats why that derivative
00:16:06.020 --> 00:16:06.580
is just 1
00:16:06.580 --> 00:16:08.000
and we end up ignoring it
00:16:08.000 --> 00:16:09.960
you dont actually have to write that
00:16:09.960 --> 00:16:11.300
term if you dont want to
00:16:11.300 --> 00:16:13.400
but you should know its sort of there
00:16:13.960 --> 00:16:15.560
in other words this is 2x
00:16:16.300 --> 00:16:18.620
dont worry well do a few more of them
00:16:28.220 --> 00:16:28.780
so its 2x
00:16:30.660 --> 00:16:31.220
6y dy/dx
00:16:32.740 --> 00:16:34.500
minus 8, 12y squared sy/sx
00:16:34.500 --> 00:16:37.300
and now comes the part where you all have problems
00:16:38.360 --> 00:16:39.480
with the algebra
00:16:40.340 --> 00:16:42.180
so you would have to isolate
00:16:42.860 --> 00:16:43.360
dy/edx
00:16:43.360 --> 00:16:44.960
this is something we did last semester
00:16:44.960 --> 00:16:46.000
to our logarithms
00:16:46.420 --> 00:16:47.940
its the same technique
00:16:49.220 --> 00:16:51.380
that you use for isolating dy/dx
00:16:52.640 --> 00:16:53.600
theres 3 steps
00:16:55.560 --> 00:16:56.920
the steps are groups
00:17:00.320 --> 00:17:01.360
factor, divide
00:17:03.800 --> 00:17:07.720
these are going to be the steps, so you look at your equation
00:17:08.780 --> 00:17:11.980
move everything that contains dy/dx on one side
00:17:11.980 --> 00:17:15.140
and everything that does contain dy/dx on the other side
00:17:15.920 --> 00:17:18.880
so on the left side ill put whatever has dy/dx
00:17:20.340 --> 00:17:20.900
6y dy/dx
00:17:22.180 --> 00:17:23.300
plus 12y squared
00:17:25.020 --> 00:17:25.520
dy/dx
00:17:25.520 --> 00:17:29.720
and on the other side i put everything that does not contain dy/dx
00:17:30.960 --> 00:17:32.080
i got 2x and i got 8
00:17:33.680 --> 00:17:34.180
8-2x
00:17:36.840 --> 00:17:38.120
so the 2 dy/dx terms
00:17:38.380 --> 00:17:39.740
stay on the left side
00:17:39.900 --> 00:17:41.740
the two terms without dy/dx
00:17:41.740 --> 00:17:43.020
go over to the other side
00:17:49.380 --> 00:17:50.900
then i factor out dy/dx
00:17:59.980 --> 00:18:01.180
so again i grouped
00:18:04.420 --> 00:18:04.980
i factor
00:18:05.920 --> 00:18:06.960
and now i divide
00:18:08.260 --> 00:18:09.540
so i get dy/dx alone
00:18:13.300 --> 00:18:14.100
equals 8-2x
00:18:15.760 --> 00:18:16.260
over 6x
00:18:17.940 --> 00:18:19.060
plus 12y squared
00:18:20.420 --> 00:18:22.180
and that is dy/dx so notice
00:18:22.340 --> 00:18:23.780
its got an x and a y in it
00:18:23.800 --> 00:18:28.200
thats why its called implicate, we dont actually know how to find y
00:18:28.520 --> 00:18:31.480
because we have x and ys mushed together here
00:18:32.380 --> 00:18:37.020
so ill do a couple more examples, dont worry i wont just leave you with 1
00:18:43.160 --> 00:18:44.160
why is it 6x?
00:18:48.400 --> 00:18:50.720
it should be 6y, just because im old
00:19:03.520 --> 00:19:07.360
lets do another one to make sure you guys have the hang of it
00:19:08.140 --> 00:19:08.640
okay so
00:19:09.280 --> 00:19:11.680
so whats going on remember you have y
00:19:11.800 --> 00:19:12.920
as a function of x
00:19:13.180 --> 00:19:15.580
so what you do the derivative of that
00:19:15.580 --> 00:19:18.740
its a chain rule you have whatever that function
00:19:19.080 --> 00:19:19.580
is
00:19:19.580 --> 00:19:22.000
and you multiply it by the derivative
00:19:23.420 --> 00:19:23.980
of y okay
00:19:24.820 --> 00:19:25.780
when you have x
00:19:25.860 --> 00:19:28.180
wed think of it as x as a function of x
00:19:28.180 --> 00:19:30.120
but that doesnt really mean anything
00:19:30.760 --> 00:19:33.320
so you would write dx/dx but dx/dx is 1`
00:19:33.320 --> 00:19:35.160
so we dont really follow the rule right
00:19:36.500 --> 00:19:38.180
lets try another example
00:19:51.980 --> 00:19:53.340
something like that
00:19:54.140 --> 00:19:56.860
you have 8x to the fourth minus 10y to the 5
00:19:57.560 --> 00:19:58.920
plus 6x cubed equals
00:20:00.080 --> 00:20:01.280
11y squared plus 4
00:20:04.440 --> 00:20:08.040
were going to take the derivative of that implicately
00:20:08.700 --> 00:20:12.220
how would you know to use the implicate depreciation
00:20:12.520 --> 00:20:13.800
youll have xs and yx
00:20:13.800 --> 00:20:16.500
or whatever youll have more than one variable in a question
00:20:16.500 --> 00:20:18.400
and you wont be able to isolate a variable
00:20:20.580 --> 00:20:21.620
cause by the way
00:20:21.620 --> 00:20:24.420
because if i didnt have this 11y squared term
00:20:24.420 --> 00:20:26.700
you could just move everything over to the other side
00:20:26.740 --> 00:20:28.260
take the fifth root of y
00:20:28.320 --> 00:20:30.080
and now you got y by its self
00:20:30.420 --> 00:20:33.860
the problem is you got 10y to the 5th plus 11y squared
00:20:33.860 --> 00:20:37.020
you cant use the qiuadratic formula if thats a fifth power
00:20:37.620 --> 00:20:41.220
by the way you wouldnt necessarily want to if i gave you
00:20:46.480 --> 00:20:47.280
you could do
00:20:48.400 --> 00:20:49.920
the quadratic formula
00:20:54.380 --> 00:20:56.620
if you take the derivative of that
00:20:56.620 --> 00:20:57.900
but ou wouldnt want to do that
00:20:58.180 --> 00:21:00.740
but its much easier to do it implicitly
00:21:03.600 --> 00:21:04.320
the idea is
00:21:04.320 --> 00:21:06.900
this is what you do when you cant isolate y
00:21:07.600 --> 00:21:09.200
so lets practice, first
00:21:09.200 --> 00:21:11.040
you take the derivative of each term
00:21:11.040 --> 00:21:13.020
the derivative of 8x to the 4th
00:21:14.320 --> 00:21:15.120
is 32x cubed
00:21:15.860 --> 00:21:17.540
times the derivative of x
00:21:18.900 --> 00:21:21.060
which is dx/dx which we cross out
00:21:21.060 --> 00:21:23.580
so were going to practice writing it a couple more times
00:21:24.480 --> 00:21:26.880
take the derivative of 10y to th e 5th
00:21:27.840 --> 00:21:28.560
which is 50
00:21:29.420 --> 00:21:30.140
y to the 4th
00:21:31.240 --> 00:21:32.920
times the derivative of y
00:21:37.020 --> 00:21:38.220
then i got 6x cubed
00:21:40.380 --> 00:21:41.740
which is 18x squared
00:21:41.920 --> 00:21:43.360
times derivative of x
00:21:50.240 --> 00:21:50.740
equals
00:21:51.600 --> 00:21:54.000
the derivative of 11y squared is 22y
00:21:55.380 --> 00:21:56.420
derivative of y
00:21:58.520 --> 00:21:59.020
plus 0
00:22:11.960 --> 00:22:12.460
group
00:22:12.460 --> 00:22:14.460
factor and divide
00:22:14.460 --> 00:22:15.020
so group
00:22:15.900 --> 00:22:18.540
everything with a y term goes on one side
00:22:18.540 --> 00:22:20.700
everything without a term goes on the other side
00:22:22.300 --> 00:22:23.580
cross out the dx/dx
00:22:26.580 --> 00:22:28.100
so i got this has a dy/dx
00:22:28.100 --> 00:22:31.500
and this has a dy/dx lets put them both on the same side
00:22:34.040 --> 00:22:35.960
and youre left with 32x cubed
00:22:38.100 --> 00:22:39.220
plus 18x squared
00:22:41.400 --> 00:22:41.900
equals
00:22:43.940 --> 00:22:44.580
22y dy/dx
00:22:50.240 --> 00:22:51.440
+50y to the 4 dy/dx
00:22:59.220 --> 00:23:01.940
everything with a dy/dx goes to the right
00:23:01.940 --> 00:23:04.300
everything without dy/dx goes to the left
00:23:05.560 --> 00:23:06.680
factor dy/dx out
00:23:10.480 --> 00:23:13.040
so you got the group, you got the factor
00:23:16.600 --> 00:23:17.400
so 32x cubed
00:23:18.040 --> 00:23:19.160
plus18x squared
00:23:20.960 --> 00:23:21.840
equals dy/dx
00:23:24.460 --> 00:23:25.100
times 22y
00:23:26.840 --> 00:23:27.880
plus 50y to the 4
00:23:30.680 --> 00:23:32.120
then divide last step
00:23:36.560 --> 00:23:38.000
and you get 32x to the 3
00:23:38.940 --> 00:23:40.060
plus 18x squared
00:23:42.080 --> 00:23:42.640
over 22y
00:23:44.400 --> 00:23:45.440
plus 50y to the 4
00:23:50.780 --> 00:23:51.820
thats not so bad
00:23:51.920 --> 00:23:55.840
youll get the hang of it well practice another one right now
00:23:58.160 --> 00:23:59.360
before i get nasty
00:24:26.180 --> 00:24:26.900
there we go
00:24:30.960 --> 00:24:32.160
y 8 minus 6y to the 5
00:24:37.180 --> 00:24:39.180
plus11x,7 +32x squared = 5x+4
00:24:39.180 --> 00:24:42.080
so you can move the x terms to the right right away
00:24:44.560 --> 00:24:45.920
you get there faster
00:24:47.020 --> 00:24:49.260
but were still stuck with dy terms
00:24:50.860 --> 00:24:53.180
derivative of y to the 8 is 8y to the 7
00:24:55.660 --> 00:24:56.160
dy/dx
00:24:59.000 --> 00:25:00.840
the derivative of 6y to the 5
00:25:01.360 --> 00:25:02.400
is 30y to the 4th
00:25:04.740 --> 00:25:05.240
dy/dx
00:25:05.240 --> 00:25:07.320
by the way some of you like y prime
00:25:07.320 --> 00:25:10.320
cuase you learn this in high school and your teacher used prime
00:25:10.320 --> 00:25:13.060
your going to want to learn to do it this way
00:25:14.140 --> 00:25:16.060
the derivative of 11x to the 7
00:25:17.440 --> 00:25:18.160
77x to the 6
00:25:18.340 --> 00:25:21.700
now theres a dx/dx there but im not going to write it
00:25:24.160 --> 00:25:26.800
then there is derivative of 32x squared
00:25:27.420 --> 00:25:28.220
which is 64x
00:25:30.060 --> 00:25:30.620
equals 5
00:25:33.640 --> 00:25:35.000
plus 0, so far so good
00:25:36.960 --> 00:25:38.320
lets do it part terms
00:25:40.820 --> 00:25:42.100
we have 8y to the 7 dy
00:25:42.720 --> 00:25:43.220
dx
00:25:44.860 --> 00:25:45.980
minus 30y to the 4
00:25:47.500 --> 00:25:48.000
dy/dx
00:25:51.600 --> 00:25:53.280
equals 5 minus 77x to the 6
00:25:55.980 --> 00:25:56.620
minus 64x
00:26:00.040 --> 00:26:00.920
so far so good
00:26:03.320 --> 00:26:04.440
factor out dy/dx
00:26:12.260 --> 00:26:12.900
8y to the 7
00:26:13.400 --> 00:26:14.680
minus 30y to the 4th
00:26:16.060 --> 00:26:18.220
equals the mess on the right side
00:26:22.820 --> 00:26:25.460
and divide im hearing very ahppy nosies
00:26:25.460 --> 00:26:28.320
this means you guys are ready for the harder problem
00:26:30.300 --> 00:26:31.020
bring it on
00:26:47.700 --> 00:26:48.580
you get 5-77x
00:26:49.040 --> 00:26:50.160
to the 6 minus 64x
00:26:51.380 --> 00:26:52.340
over 8y to the y
00:26:52.960 --> 00:26:54.080
minus 30y to the 4
00:26:54.360 --> 00:26:56.440
okay how can i make this harder?
00:27:13.120 --> 00:27:15.520
so how could i make this harder? well
00:27:15.540 --> 00:27:17.540
how bout i give you one of these
00:27:29.840 --> 00:27:30.560
find dy/dx
00:27:31.240 --> 00:27:33.640
youre going to need the product rule
00:27:35.640 --> 00:27:37.480
so lets take the derivative
00:27:37.480 --> 00:27:41.920
so first we have to take the derivative of 5x cubed, thats 15x squared
00:27:42.440 --> 00:27:46.200
times the derivative of x and the derivative of x is just 1
00:27:46.220 --> 00:27:48.540
so ignore that, now for the fun part
00:27:49.340 --> 00:27:51.740
product rule, so we have 2 functions
00:27:52.700 --> 00:27:55.260
we have 4x squared and we have y squared
00:27:55.500 --> 00:27:58.380
so first lets leave the four x squared alone
00:28:00.460 --> 00:28:03.420
times the derivative of y squared which is 2y
00:28:06.760 --> 00:28:07.260
dy/dx
00:28:08.960 --> 00:28:11.760
plus reverse because this is product rule
00:28:13.000 --> 00:28:15.000
derivative of 4x squared is 8x
00:28:15.000 --> 00:28:17.200
times the derivative of x which is 1
00:28:18.900 --> 00:28:19.940
times y squared
00:28:22.280 --> 00:28:24.040
thats the middle part plus
00:28:25.580 --> 00:28:27.340
the derivative of 6y cubed
00:28:28.160 --> 00:28:28.960
18y squared
00:28:30.180 --> 00:28:30.680
dy/dx
00:28:32.540 --> 00:28:33.100
equals 0
00:28:34.300 --> 00:28:39.260
now suppose this problem was find the tangent line of an equation like this
00:28:41.120 --> 00:28:42.320
at x=1 y= whatever
00:28:42.880 --> 00:28:44.480
you immedaitely plug in
00:28:44.480 --> 00:28:48.860
you dont want to do the algebra of isolating dy/dx thats a lot of work
00:28:48.860 --> 00:28:52.680
theres a chance youll mess it up algebra can be a bit tricky
00:28:53.120 --> 00:28:57.040
but however we say find dy/dx now you have to do all the steps
00:28:57.040 --> 00:29:00.160
lets do this again, make sure you understood what i did
00:29:00.280 --> 00:29:03.000
the derivative of 5x cubed is 15x squared
00:29:03.020 --> 00:29:06.460
and 4x squared y squared you have two functions here
00:29:06.460 --> 00:29:08.040
you have 4x squared, y squared
00:29:08.960 --> 00:29:10.240
s its four x squared
00:29:11.240 --> 00:29:13.320
times y squared which is 2dy/dx
00:29:14.160 --> 00:29:16.880
plus derivative of 4x squared which is 8x
00:29:16.880 --> 00:29:18.380
times the derivative of y squared
00:29:19.460 --> 00:29:22.660
taking the derivative of 6y cubed is 18y squared
00:29:23.340 --> 00:29:23.840
dy/dx
00:29:24.680 --> 00:29:25.720
derovative of 1 is just 0
00:29:26.440 --> 00:29:31.400
so before we do our technique which have to do some simplifying so 15x cubed
00:29:34.340 --> 00:29:36.820
and you get lets see 8x squaredy dy/dx
00:29:41.400 --> 00:29:42.520
plus 8xy squared
00:29:44.300 --> 00:29:44.860
plus 18y
00:29:46.540 --> 00:29:47.500
squared dy/dx
00:29:54.500 --> 00:29:56.100
now group factor divide
00:29:57.380 --> 00:30:01.700
so now lets put everything with dy/dx and leave it on the left side
00:30:01.700 --> 00:30:03.520
more the other term to the right side
00:30:08.580 --> 00:30:09.860
so i get 8x squared y
00:30:10.840 --> 00:30:11.340
dy/dx
00:30:13.740 --> 00:30:14.860
plus 18y squared
00:30:16.200 --> 00:30:16.700
dy/dx
00:30:20.120 --> 00:30:21.480
equals -15x squared
00:30:24.780 --> 00:30:25.980
minus 8xy squared
00:30:31.220 --> 00:30:32.340
factor out dy/dx
00:30:49.120 --> 00:30:49.840
and divide
00:31:00.020 --> 00:31:00.900
-15x squared
00:31:02.440 --> 00:31:03.320
-8xy squared
00:31:06.860 --> 00:31:07.980
over 8x squared y
00:31:09.240 --> 00:31:10.360
plus 18y squared
00:31:13.960 --> 00:31:15.240
lets do another one
00:31:15.240 --> 00:31:17.740
how else would i make this really hard, yes
00:31:29.140 --> 00:31:30.580
why isnt ther a dy term
00:31:31.100 --> 00:31:34.460
when i took the derivative of y squared i got 2 dy/dx
00:31:36.140 --> 00:31:37.900
now put in the product rule
00:31:39.720 --> 00:31:42.040
times the derivative of x which is 1
00:31:43.960 --> 00:31:47.560
the derivative of the 4x squared doesnt contain any ys
00:31:47.580 --> 00:31:49.420
so thats why theres no dy/dx
00:31:50.820 --> 00:31:54.740
remember im doing the derivative of 2 different functions
00:32:05.600 --> 00:32:09.440
so one more time, if i just had the derivative of 4x squared
00:32:11.260 --> 00:32:14.140
y squared, its the derivative of 4x squared
00:32:14.340 --> 00:32:16.580
timesthe derivative of y squared
00:32:17.540 --> 00:32:18.040
dy/dx
00:32:19.660 --> 00:32:20.620
plus y squared
00:32:20.620 --> 00:32:22.360
times the derivative of 4x
00:32:24.560 --> 00:32:26.880
theres no dy/dx because its just 8x
00:32:26.880 --> 00:32:28.820
times the derivative of x which is 1
00:32:31.960 --> 00:32:34.680
this is tricky you have to do it a few times
00:32:34.840 --> 00:32:37.160
thats why ill put more stuff online
00:32:37.160 --> 00:32:40.140
after you do 3 or 4 thousand of them youll get good at them
00:32:52.840 --> 00:32:55.400
could i make this more annoying oh sure
00:33:04.380 --> 00:33:05.420
lets find dy/dx
00:33:08.880 --> 00:33:11.840
the derivative of sin, the outside function
00:33:12.160 --> 00:33:12.660
is sin
00:33:13.640 --> 00:33:15.640
you get cosxy, you get nothing
00:33:18.400 --> 00:33:20.160
times the derivative of xy
00:33:20.160 --> 00:33:22.160
thats going to require the product rule
00:33:24.680 --> 00:33:27.400
x times the derivative of y which is dy/dx
00:33:32.280 --> 00:33:34.360
plus y times the derivative of x
00:33:38.280 --> 00:33:39.640
thats your left side
00:33:42.240 --> 00:33:43.760
the right side is 1 plus
00:33:44.920 --> 00:33:45.420
dy/dx
00:33:46.280 --> 00:33:47.400
that wasnt so bad
00:33:49.820 --> 00:33:51.100
were not done, oh no
00:33:53.900 --> 00:33:56.540
we have to isolate dy/dx how do we do that
00:33:57.820 --> 00:34:00.540
you dont divide like cos, you distribute
00:34:01.520 --> 00:34:03.040
so first you take this x
00:34:05.040 --> 00:34:05.760
cos x dy/dx
00:34:09.320 --> 00:34:09.820
plus y
00:34:11.720 --> 00:34:12.220
cosx y
00:34:15.380 --> 00:34:16.260
equals 1 plus
00:34:18.980 --> 00:34:19.480
dy dx
00:34:24.940 --> 00:34:28.460
now the same as before move the dy/dx terms to one side
00:34:28.580 --> 00:34:30.500
and the non dy/dx to the other
00:34:30.920 --> 00:34:32.840
so i put y/dx to the left i get x
00:34:34.860 --> 00:34:35.360
cosxy
00:34:36.620 --> 00:34:37.120
dy/dx
00:34:39.220 --> 00:34:40.020
minus dy/dx
00:34:43.800 --> 00:34:44.520
equals 1-y
00:34:46.020 --> 00:34:46.520
cosxy
00:34:57.000 --> 00:34:58.280
next pull out dy/dx
00:35:04.800 --> 00:35:05.300
xcoxy
00:35:07.120 --> 00:35:07.620
minus 1
00:35:10.120 --> 00:35:11.160
equals 1-cosxy
00:35:17.880 --> 00:35:19.480
and divide you get dy/dx
00:35:24.620 --> 00:35:25.120
1-y
00:35:26.480 --> 00:35:26.980
cosxy
00:35:30.860 --> 00:35:31.660
over xcosxy
00:35:32.920 --> 00:35:37.960
minus 1, normally i could ask for the second derivative but i wouldnt do that
00:35:37.960 --> 00:35:40.800
remember what i told you the very first class
00:35:40.800 --> 00:35:44.420
the problem with clac isnt the derivative part its the calculus part
00:35:46.120 --> 00:35:48.040
this first step isnt so rough
00:35:48.040 --> 00:35:51.300
its the remaining part that gets you through that messes you up
00:35:51.780 --> 00:35:57.140
so thats what you practice, remember what i said ill put some stuff on blackboard