1
00:00:00,000 --> 00:00:07,440

2
00:00:07,440 --> 00:00:09,600
PROFESSOR: Welcome back
to recitation.

3
00:00:09,600 --> 00:00:13,230
In this segment I'm going to
actually show, well, you're

4
00:00:13,230 --> 00:00:16,250
actually going to show, the
derivative of tangent x using

5
00:00:16,250 --> 00:00:17,300
the quotient rule.

6
00:00:17,300 --> 00:00:20,670
So what I'd like you to do, I
wanted to remind you of what

7
00:00:20,670 --> 00:00:21,810
the quotient rule is.

8
00:00:21,810 --> 00:00:23,750
So u and v are functions of x.

9
00:00:23,750 --> 00:00:27,200
We want to take the derivative
of u divided by v. I've

10
00:00:27,200 --> 00:00:31,420
written the formula that you
were given in class for this.

11
00:00:31,420 --> 00:00:36,280
And I'm asking for you to take
d dx of tangent x using the

12
00:00:36,280 --> 00:00:37,240
quotient rule.

13
00:00:37,240 --> 00:00:39,810
And the hint I will give you is
the reason we can obviously

14
00:00:39,810 --> 00:00:43,130
use the quotient rule is because
tangent x is equal to

15
00:00:43,130 --> 00:00:44,640
a quotient of two
functions of x.

16
00:00:44,640 --> 00:00:47,050
It's sine x divided
by cosine of x.

17
00:00:47,050 --> 00:00:49,330
So I'm going to give you a
minute to work this out for

18
00:00:49,330 --> 00:00:51,040
yourself, and then when
we come back I

19
00:00:51,040 --> 00:00:52,390
will do it for you.

20
00:00:52,390 --> 00:00:55,170

21
00:00:55,170 --> 00:00:55,500
OK.

22
00:00:55,500 --> 00:00:58,220
So we want to find the
derivative of tangent x.

23
00:00:58,220 --> 00:01:02,200
So let me, let me work on
this side of the board.

24
00:01:02,200 --> 00:01:09,500
So I'm actually going to take
d dx of sine x divided by

25
00:01:09,500 --> 00:01:11,710
cosine of x.

26
00:01:11,710 --> 00:01:11,980
OK.

27
00:01:11,980 --> 00:01:15,160
So in this case, sign x is u.

28
00:01:15,160 --> 00:01:20,280
Cosine x is v. So using my
quotient rule I know that

29
00:01:20,280 --> 00:01:23,080
first I have to take the
derivative of sine x--

30
00:01:23,080 --> 00:01:25,980
that's cosine x-- and then
I multiply it by the

31
00:01:25,980 --> 00:01:28,000
denominator, the v,
which is cosine x.

32
00:01:28,000 --> 00:01:33,320
So my first term in the
numerator is cosine squared x.

33
00:01:33,320 --> 00:01:37,520
Again, one cosine x comes from
the derivative of sine x, one

34
00:01:37,520 --> 00:01:41,630
cosine x is the v. It's the
cosine x in the denominator.

35
00:01:41,630 --> 00:01:45,250
Then I have to subtract
v prime u.

36
00:01:45,250 --> 00:01:48,270
The derivative of cosine
x is negative sine x.

37
00:01:48,270 --> 00:01:52,710
I'll actually just write
that one down.

38
00:01:52,710 --> 00:01:55,520
And then I bring the u
along for the ride.

39
00:01:55,520 --> 00:01:57,580
So I multiply by sine x here.

40
00:01:57,580 --> 00:02:00,120

41
00:02:00,120 --> 00:02:02,260
And then I take v
squared in the

42
00:02:02,260 --> 00:02:03,460
denominator from the formala.

43
00:02:03,460 --> 00:02:07,290
v, again, is cosine x, so I take
cosine squared x in the

44
00:02:07,290 --> 00:02:08,360
denominator.

45
00:02:08,360 --> 00:02:10,920
Now this at this point is a
little bit messy, but the nice

46
00:02:10,920 --> 00:02:13,670
thing is that we can use some
trigonometric identities to

47
00:02:13,670 --> 00:02:14,820
simplify this.

48
00:02:14,820 --> 00:02:19,820
So let me first write out what
it is a little more clearly.

49
00:02:19,820 --> 00:02:22,770
Minus a negative gives
you a positive.

50
00:02:22,770 --> 00:02:26,200
And then here I get sine x times
sine x, so I get the

51
00:02:26,200 --> 00:02:28,700
sine squared x.

52
00:02:28,700 --> 00:02:33,320
And then I keep divided
by cosine squared x.

53
00:02:33,320 --> 00:02:37,160
Now at this point some of you
might have divided by cosine

54
00:02:37,160 --> 00:02:40,730
squared x here and gotten 1, and
divided by cosine squared

55
00:02:40,730 --> 00:02:42,890
x here and gotten tangent
squared x.

56
00:02:42,890 --> 00:02:46,030
And then from there you could
simplify to another

57
00:02:46,030 --> 00:02:47,430
trigonometric function.

58
00:02:47,430 --> 00:02:50,490
I'm going to go straight a
different way to show you what

59
00:02:50,490 --> 00:02:52,140
that actually also equals.

60
00:02:52,140 --> 00:02:54,540
So there, at this point I want
to stress there are sort of

61
00:02:54,540 --> 00:02:57,640
two ways you can get
to the same place.

62
00:02:57,640 --> 00:03:00,230
But I'm going to use the fact
that the numerator is a very

63
00:03:00,230 --> 00:03:02,950
nice trigonometric identity
that we know.

64
00:03:02,950 --> 00:03:05,560
We know cosine squared
x plus sine squared x

65
00:03:05,560 --> 00:03:06,890
always equals 1.

66
00:03:06,890 --> 00:03:11,610
So this is quite lovely, the
numerator simplifies to 1, the

67
00:03:11,610 --> 00:03:15,460
denominator stays cosine
squared x.

68
00:03:15,460 --> 00:03:16,980
What is this function?

69
00:03:16,980 --> 00:03:20,070
1 over cosine x is actually
secant x.

70
00:03:20,070 --> 00:03:25,260
So if you need at this point
to rewrite the whole thing

71
00:03:25,260 --> 00:03:27,180
like this, right?--

72
00:03:27,180 --> 00:03:29,610
1 squared is 1 and in the
denominator we still get

73
00:03:29,610 --> 00:03:30,970
cosine squared x--

74
00:03:30,970 --> 00:03:35,680
this tells you that 1 over
cosine squared x is actually

75
00:03:35,680 --> 00:03:40,100
just equal to secant
squared x.

76
00:03:40,100 --> 00:03:42,630
So again, what I want to point
out is we've now taken the

77
00:03:42,630 --> 00:03:45,340
derivative of tangent x
and we got that that's

78
00:03:45,340 --> 00:03:47,400
secant squared x.

79
00:03:47,400 --> 00:03:50,970
Now using this quotient rule,
you can do the same kind of

80
00:03:50,970 --> 00:03:54,760
thing with cotangent x, with
cosecant x, with secant x.

81
00:03:54,760 --> 00:03:56,590
You can find all these
derivatives of these

82
00:03:56,590 --> 00:03:59,750
trigonometric functions using
the quotient rule.

83
00:03:59,750 --> 00:04:03,250
So if you want to know what the
derivative of secant x is,

84
00:04:03,250 --> 00:04:06,640
you should take dd x of 1
divided by cosine x and use

85
00:04:06,640 --> 00:04:07,710
the quotient rule.

86
00:04:07,710 --> 00:04:11,960
Or, in fact, the chain rule
would work well there also, to

87
00:04:11,960 --> 00:04:12,780
find that derivative.

88
00:04:12,780 --> 00:04:16,430
So we are building up the number
of derivatives we can

89
00:04:16,430 --> 00:04:18,540
find using these different
rules.

90
00:04:18,540 --> 00:04:20,850
So we'll stop there.

91
00:04:20,850 --> 00:04:21,261