1
00:00:14,500 --> 00:00:18,652
In the last videos I talked about the derivatives of simple functions,

2
00:00:18,652 --> 00:00:22,455
and the goal was to have a clear picture or intuition to hold in

3
00:00:22,455 --> 00:00:26,199
your mind that actually explains where these formulas come from.

4
00:00:26,839 --> 00:00:31,387
But most of the functions you deal with in modeling the world involve mixing,

5
00:00:31,387 --> 00:00:35,176
combining, or tweaking these simple functions in some other way,

6
00:00:35,176 --> 00:00:39,782
so our next step is to understand how you take derivatives of more complicated

7
00:00:39,781 --> 00:00:40,539
combinations.

8
00:00:41,280 --> 00:00:43,995
Again, I don't want these to be something to memorize,

9
00:00:43,994 --> 00:00:47,599
I want you to have a clear picture in mind for where each one comes from.

10
00:00:49,520 --> 00:00:53,600
Now, this really boils down into three basic ways to combine functions.

11
00:00:54,100 --> 00:00:56,591
You can add them together, you can multiply them,

12
00:00:56,591 --> 00:00:59,780
and you can throw one inside the other, known as composing them.

13
00:01:00,600 --> 00:01:03,829
Sure, you could say subtracting them, but really that's just

14
00:01:03,829 --> 00:01:07,219
multiplying the second by negative one and adding them together.

15
00:01:08,239 --> 00:01:11,164
Likewise, dividing functions doesn't really add anything,

16
00:01:11,164 --> 00:01:14,844
because that's the same as plugging one inside the function, one over x,

17
00:01:14,843 --> 00:01:16,759
and then multiplying the two together.

18
00:01:17,659 --> 00:01:20,664
So really, most functions you come across just involve layering

19
00:01:20,664 --> 00:01:23,200
together these three different types of combinations,

20
00:01:23,200 --> 00:01:26,439
though there's not really a bound on how monstrous things can become.

21
00:01:27,099 --> 00:01:31,298
But as long as you know how derivatives play with just those three combination types,

22
00:01:31,299 --> 00:01:34,376
you'll always be able to take it step by step and peel through

23
00:01:34,376 --> 00:01:36,719
the layers for any kind of monstrous expression.

24
00:01:38,719 --> 00:01:42,540
So the question is, if you know the derivative of two functions,

25
00:01:42,540 --> 00:01:45,774
what is the derivative of their sum, of their product,

26
00:01:45,774 --> 00:01:48,420
and of the function composition between them?

27
00:01:50,319 --> 00:01:54,259
The sum rule is easiest, if somewhat tongue-twisting to say out loud.

28
00:01:54,840 --> 00:01:58,600
The derivative of a sum of two functions is the sum of their derivatives.

29
00:01:59,799 --> 00:02:03,700
But it's worth warming up with this example by really thinking through

30
00:02:03,700 --> 00:02:07,105
what it means to take a derivative of a sum of two functions,

31
00:02:07,105 --> 00:02:11,170
since the derivative patterns for products and function composition won't

32
00:02:11,169 --> 00:02:15,619
be so straightforward, and they're going to require this kind of deeper thinking.

33
00:02:16,699 --> 00:02:21,199
For example, let's think about this function f of x equals sine of x plus x squared.

34
00:02:22,199 --> 00:02:25,211
It's a function where, for every input, you add together

35
00:02:25,211 --> 00:02:27,959
the values of sine of x and x squared at that point.

36
00:02:29,759 --> 00:02:34,048
For example, let's say at x equals 0.5, the height of the sine

37
00:02:34,049 --> 00:02:38,201
graph is given by this vertical bar, and the height of the x

38
00:02:38,201 --> 00:02:42,560
squared parabola is given by this slightly smaller vertical bar.

39
00:02:44,379 --> 00:02:47,319
And their sum is the length you get by just stacking them together.

40
00:02:48,520 --> 00:02:53,939
For the derivative, you want to ask what happens as you nudge that input slightly,

41
00:02:53,938 --> 00:02:56,419
maybe increasing it up to 0.5 plus dx.

42
00:02:57,560 --> 00:03:02,920
The difference in the value of f between those two places is what we call df.

43
00:03:04,360 --> 00:03:08,978
And when you picture it like this, I think you'll agree that the total

44
00:03:08,978 --> 00:03:13,271
change in the height is whatever the change to the sine graph is,

45
00:03:13,270 --> 00:03:18,799
what we might call d sine of x, plus whatever the change to x squared is, dx squared.

46
00:03:22,240 --> 00:03:27,540
We know that the derivative of sine is cosine, and remember what that means.

47
00:03:27,919 --> 00:03:33,299
It means that this little change, d sine of x, is about cosine of x times dx.

48
00:03:33,780 --> 00:03:37,711
It's proportional to the size of our initial nudge dx,

49
00:03:37,711 --> 00:03:43,359
and the proportionality constant equals cosine of whatever input we started at.

50
00:03:43,979 --> 00:03:48,072
Likewise, because the derivative of x squared is 2x,

51
00:03:48,072 --> 00:03:53,939
the change in the height of the x squared graph is 2x times whatever dx was.

52
00:03:55,599 --> 00:04:00,475
So rearranging df divided by dx, the ratio of the tiny change to

53
00:04:00,475 --> 00:04:04,752
the sum function to the tiny change in x that caused it,

54
00:04:04,752 --> 00:04:10,079
is indeed cosine of x plus 2x, the sum of the derivatives of its parts.

55
00:04:11,520 --> 00:04:15,329
But like I said, things are a bit different for products,

56
00:04:15,329 --> 00:04:19,139
and let's think through why in terms of tiny nudges again.

57
00:04:20,060 --> 00:04:23,139
In this case, I don't think graphs are our best bet for visualizing things.

58
00:04:23,819 --> 00:04:27,040
Pretty commonly in math, at a lot of levels of math really,

59
00:04:27,040 --> 00:04:29,617
if you're dealing with a product of two things,

60
00:04:29,617 --> 00:04:32,140
it helps to understand it as some kind of area.

61
00:04:33,079 --> 00:04:36,014
In this case, maybe you try to configure some mental setup

62
00:04:36,014 --> 00:04:39,000
of a box where the side lengths are sine of x and x squared.

63
00:04:39,879 --> 00:04:41,040
But what would that mean?

64
00:04:42,319 --> 00:04:46,606
Well, since these are functions, you might think of those sides as adjustable,

65
00:04:46,607 --> 00:04:49,972
dependent on the value of x, which maybe you think of as this

66
00:04:49,971 --> 00:04:52,739
number that you can just freely adjust up and down.

67
00:04:53,740 --> 00:04:56,812
So getting a feel for what this means, focus on

68
00:04:56,812 --> 00:05:00,139
that top side who changes as the function sine of x.

69
00:05:01,060 --> 00:05:05,305
As you change this value of x up from 0, it increases up to

70
00:05:05,305 --> 00:05:09,127
a length of 1 as sine of x moves up towards its peak,

71
00:05:09,127 --> 00:05:13,939
and after that it starts to decrease as sine of x comes down from 1.

72
00:05:15,100 --> 00:05:18,580
And in the same way, that height there is always changing as x squared.

73
00:05:20,079 --> 00:05:25,800
So f of x, defined as the product of these two functions, is the area of this box.

74
00:05:27,060 --> 00:05:30,120
And for the derivative, let's think about how

75
00:05:30,120 --> 00:05:33,180
a tiny change to x by dx influences that area.

76
00:05:33,839 --> 00:05:36,279
What is that resulting change in area df?

77
00:05:39,000 --> 00:05:44,115
Well, the nudge dx caused that width to change by some small d sine of x,

78
00:05:44,115 --> 00:05:47,919
and it caused that height to change by some dx squared.

79
00:05:50,180 --> 00:05:53,649
And this gives us three little snippets of new area,

80
00:05:53,649 --> 00:05:58,033
a thin rectangle on the bottom whose area is its width, sine of x,

81
00:05:58,033 --> 00:06:00,259
times its thin height, dx squared.

82
00:06:01,779 --> 00:06:06,406
And there's this thin rectangle on the right, whose area is its height,

83
00:06:06,406 --> 00:06:09,299
x squared, times its thin width, d sine of x.

84
00:06:10,740 --> 00:06:14,139
And there's also this little bit in the corner, but we can ignore that.

85
00:06:14,439 --> 00:06:17,856
Its area is ultimately proportional to dx squared,

86
00:06:17,857 --> 00:06:22,480
and as we've seen before, that becomes negligible as dx goes to zero.

87
00:06:23,939 --> 00:06:27,375
I mean, this whole setup is very similar to what I showed last video,

88
00:06:27,375 --> 00:06:28,699
with the x squared diagram.

89
00:06:29,459 --> 00:06:32,704
And just like then, keep in mind that I'm using somewhat beefy

90
00:06:32,704 --> 00:06:35,899
changes here to draw things, just so we can actually see them.

91
00:06:36,360 --> 00:06:39,818
But in principle, dx is something very very small,

92
00:06:39,817 --> 00:06:44,699
and that means that dx squared and d sine of x are also very very small.

93
00:06:45,980 --> 00:06:51,175
So, applying what we know about the derivative of sine and of x squared,

94
00:06:51,175 --> 00:06:55,660
that tiny change, dx squared, is going to be about 2x times dx.

95
00:06:56,360 --> 00:07:01,580
And that tiny change, d sine of x, well that's going to be about cosine of x times dx.

96
00:07:02,920 --> 00:07:09,019
As usual, we divide out by that dx to see that the ratio we want, df divided by dx,

97
00:07:09,019 --> 00:07:12,504
is sine of x times the derivative of x squared,

98
00:07:12,504 --> 00:07:15,699
plus x squared times the derivative of sine.

99
00:07:17,959 --> 00:07:21,259
And nothing we've done here is specific to sine or to x squared.

100
00:07:21,579 --> 00:07:25,359
This same line of reasoning would work for any two functions, g and h.

101
00:07:27,000 --> 00:07:29,310
And sometimes people like to remember this pattern with

102
00:07:29,310 --> 00:07:31,539
a certain mnemonic that you kind of sing in your head.

103
00:07:32,220 --> 00:07:33,680
Left d right, right d left.

104
00:07:34,399 --> 00:07:38,812
In this example, where we have sine of x times x squared, left d right,

105
00:07:38,812 --> 00:07:43,778
means you take that left function, sine of x, times the derivative of the right,

106
00:07:43,778 --> 00:07:44,759
in this case 2x.

107
00:07:45,480 --> 00:07:48,876
Then you add on right d left, that right function,

108
00:07:48,875 --> 00:07:52,939
x squared, times the derivative of the left one, cosine of x.

109
00:07:54,360 --> 00:07:57,271
Now out of context, presented as a rule to remember,

110
00:07:57,271 --> 00:08:00,019
I think this would feel pretty strange, don't you?

111
00:08:00,740 --> 00:08:03,381
But when you actually think of this adjustable box,

112
00:08:03,380 --> 00:08:05,819
you can see what each of those terms represents.

113
00:08:06,579 --> 00:08:10,971
Left d right is the area of that little bottom rectangle,

114
00:08:10,971 --> 00:08:15,439
and right d left is the area of that rectangle on the side.

115
00:08:20,160 --> 00:08:23,847
By the way, I should mention that if you multiply by a constant,

116
00:08:23,846 --> 00:08:26,739
say 2 times sine of x, things end up a lot simpler.

117
00:08:27,399 --> 00:08:30,875
The derivative is just the same as the constant multiplied by

118
00:08:30,875 --> 00:08:34,519
the derivative of the function, in this case 2 times cosine of x.

119
00:08:35,558 --> 00:08:40,178
I'll leave it to you to pause and ponder and verify that makes sense.

120
00:08:41,918 --> 00:08:46,533
Aside from addition and multiplication, the other common way to combine functions,

121
00:08:46,533 --> 00:08:49,201
and believe me, this one comes up all the time,

122
00:08:49,201 --> 00:08:52,259
is to shove one inside the other, function composition.

123
00:08:53,220 --> 00:08:56,898
For example, maybe we take the function x squared and shove it

124
00:08:56,898 --> 00:09:00,460
inside sine of x to get this new function, sine of x squared.

125
00:09:01,399 --> 00:09:04,079
What do you think the derivative of that new function is?

126
00:09:05,299 --> 00:09:09,146
To think this one through, I'll choose yet another way to visualize things,

127
00:09:09,147 --> 00:09:12,540
just to emphasize that in creative math, we've got lots of options.

128
00:09:13,320 --> 00:09:18,591
I'll put up three different number lines, the top one is going to hold the value of x,

129
00:09:18,591 --> 00:09:21,440
the second one is going to hold the x squared,

130
00:09:21,440 --> 00:09:25,500
and the third line is going to hold the value of sine of x squared.

131
00:09:26,460 --> 00:09:30,310
That is, the function x squared gets you from line 1 to line 2,

132
00:09:30,309 --> 00:09:33,500
and the function sine gets you from line 2 to line 3.

133
00:09:34,840 --> 00:09:39,581
As I shift around this value of x, maybe moving it up to the value 3,

134
00:09:39,581 --> 00:09:45,340
that second value stays pegged to whatever x squared is, in this case moving up to 9.

135
00:09:46,200 --> 00:09:49,355
That bottom value, being sine of x squared, is

136
00:09:49,355 --> 00:09:52,580
going to go to whatever sine of 9 happens to be.

137
00:09:54,899 --> 00:10:00,399
So, for the derivative, let's again start by nudging that x value by some little dx.

138
00:10:01,539 --> 00:10:04,799
I always think that it's helpful to think of x as starting

139
00:10:04,799 --> 00:10:07,839
at some actual concrete number, maybe 1.5 in this case.

140
00:10:08,759 --> 00:10:14,737
The resulting nudge to that second value, the change in x squared caused by such a dx,

141
00:10:14,738 --> 00:10:15,700
is dx squared.

142
00:10:16,960 --> 00:10:21,062
We could expand this like we have before, as 2x times dx,

143
00:10:21,062 --> 00:10:25,307
which for our specific input would be 2 times 1.5 times dx,

144
00:10:25,307 --> 00:10:30,120
but it helps to keep things written as dx squared, at least for now.

145
00:10:31,019 --> 00:10:36,384
In fact, I'm going to go one step further, give a new name to this x squared,

146
00:10:36,384 --> 00:10:41,200
maybe h, so instead of writing dx squared for this nudge, we write dh.

147
00:10:42,620 --> 00:10:47,259
This makes it easier to think about that third value, which is now pegged at sine of h.

148
00:10:48,200 --> 00:10:53,680
Its change is d sine of h, the tiny change caused by the nudge dh.

149
00:10:55,000 --> 00:11:00,134
By the way, the fact that it's moving to the left while the dh bump is going to the right

150
00:11:00,134 --> 00:11:05,039
just means that this change, d sine of h, is going to be some kind of negative number.

151
00:11:06,139 --> 00:11:09,639
Once again, we can use our knowledge of the derivative of the sine.

152
00:11:10,500 --> 00:11:14,419
This d sine of h is going to be about cosine of h times dh.

153
00:11:15,240 --> 00:11:18,639
That's what it means for the derivative of sine to be cosine.

154
00:11:19,539 --> 00:11:23,771
Unfolding things, we can replace that h with x squared again,

155
00:11:23,772 --> 00:11:29,780
so we know that the bottom nudge will be a size of cosine of x squared times dx squared.

156
00:11:31,039 --> 00:11:32,480
Let's unfold things even further.

157
00:11:32,840 --> 00:11:38,100
That intermediate nudge dx squared is going to be about 2x times dx.

158
00:11:39,059 --> 00:11:41,445
It's always a good habit to remind yourself of

159
00:11:41,446 --> 00:11:43,680
what an expression like this actually means.

160
00:11:44,340 --> 00:11:48,578
In this case, where we started at x equals 1.5 up top,

161
00:11:48,577 --> 00:11:54,512
this whole expression is telling us that the size of the nudge on that third

162
00:11:54,513 --> 00:12:00,754
line is going to be about cosine of 1.5 squared times 2 times 1.5 times whatever

163
00:12:00,754 --> 00:12:02,220
the size of dx was.

164
00:12:02,720 --> 00:12:05,112
It's proportional to the size of dx, and this

165
00:12:05,111 --> 00:12:07,919
derivative is giving us that proportionality constant.

166
00:12:10,919 --> 00:12:12,559
Notice what we came out with here.

167
00:12:12,960 --> 00:12:15,855
We have the derivative of the outside function,

168
00:12:15,855 --> 00:12:19,235
and it's still taking in the unaltered inside function,

169
00:12:19,235 --> 00:12:23,220
and then multiplying it by the derivative of that inside function.

170
00:12:25,820 --> 00:12:29,220
Again, there's nothing special about sine of x or x squared.

171
00:12:29,740 --> 00:12:33,501
If you have any two functions, g of x and h of x,

172
00:12:33,501 --> 00:12:37,263
the derivative of their composition, g of h of x,

173
00:12:37,264 --> 00:12:43,659
is going to be the derivative of g evaluated on h, multiplied by the derivative of h.

174
00:12:47,139 --> 00:12:50,899
This pattern right here is what we usually call the chain rule.

175
00:12:52,039 --> 00:12:57,679
Notice for the derivative of g, I'm writing it as dg dh instead of dg dx.

176
00:12:58,679 --> 00:13:02,234
On the symbolic level, this is a reminder that the thing you plug

177
00:13:02,235 --> 00:13:06,060
into that derivative is still going to be that intermediary function h.

178
00:13:07,019 --> 00:13:09,745
But more than that, it's an important reflection of what

179
00:13:09,745 --> 00:13:12,519
this derivative of the outer function actually represents.

180
00:13:13,200 --> 00:13:18,449
Remember, in our three line setup, when we took the derivative of the sine on

181
00:13:18,448 --> 00:13:23,899
that bottom, we expanded the size of that nudge, d sine, as cosine of h times dh.

182
00:13:24,940 --> 00:13:27,496
This was because we didn't immediately know how

183
00:13:27,495 --> 00:13:29,840
the size of that bottom nudge depended on x.

184
00:13:30,419 --> 00:13:32,599
That's kind of the whole thing we were trying to figure out.

185
00:13:33,259 --> 00:13:37,360
But we could take the derivative with respect to that intermediate variable, h.

186
00:13:38,100 --> 00:13:41,916
That is, figure out how to express the size of that nudge on the third

187
00:13:41,916 --> 00:13:45,680
line as some multiple of dh, the size of the nudge on the second line.

188
00:13:46,580 --> 00:13:50,700
It was only after that that we unfolded further by figuring out what dh was.

189
00:13:53,320 --> 00:13:58,727
In this chain rule expression, we're saying, look at the ratio between a tiny change in

190
00:13:58,726 --> 00:14:02,351
g, the final output, to a tiny change in h that caused it,

191
00:14:02,351 --> 00:14:04,379
h being the value we plug into g.

192
00:14:05,320 --> 00:14:08,680
Then multiply that by the tiny change in h, divided

193
00:14:08,679 --> 00:14:11,199
by the tiny change in x that caused it.

194
00:14:12,299 --> 00:14:15,481
So notice, those dh's cancel out, and they give us a ratio

195
00:14:15,481 --> 00:14:19,473
between the change in that final output and the change to the input that,

196
00:14:19,474 --> 00:14:22,280
through a certain chain of events, brought it about.

197
00:14:23,860 --> 00:14:26,980
And that cancellation of dh is not just a notational trick.

198
00:14:26,980 --> 00:14:30,350
That is a genuine reflection of what's going on with the

199
00:14:30,350 --> 00:14:33,899
tiny nudges that underpin everything we do with derivatives.

200
00:14:36,299 --> 00:14:39,877
So those are the three basic tools to have in your belt to handle

201
00:14:39,878 --> 00:14:43,240
derivatives of functions that combine a lot of smaller things.

202
00:14:43,840 --> 00:14:47,379
You've got the sum rule, the product rule, and the chain rule.

203
00:14:48,399 --> 00:14:51,922
And I'll be honest with you, there is a big difference between knowing

204
00:14:51,922 --> 00:14:54,551
what the chain rule is and what the product rule is,

205
00:14:54,551 --> 00:14:58,620
and actually being fluent with applying them in even the most hairy of situations.

206
00:14:59,480 --> 00:15:03,100
Watching videos, any videos, about the mechanics of calculus is

207
00:15:03,100 --> 00:15:06,892
never going to substitute for practicing those mechanics yourself,

208
00:15:06,892 --> 00:15:10,400
and building up the muscles to do these computations yourself.

209
00:15:11,240 --> 00:15:13,600
I really wish I could offer to do that for you,

210
00:15:13,600 --> 00:15:17,440
but I'm afraid the ball is in your court, my friend, to seek out the practice.

211
00:15:18,039 --> 00:15:20,940
What I can offer, and what I hope I have offered,

212
00:15:20,941 --> 00:15:23,960
is to show you where these rules actually come from.

213
00:15:24,139 --> 00:15:27,774
To show that they're not just something to be memorized and hammered away,

214
00:15:27,774 --> 00:15:31,264
but they're natural patterns, things that you too could have discovered

215
00:15:31,264 --> 00:15:34,560
just by patiently thinking through what a derivative actually means.