1
00:00:02,775 --> 00:00:05,105
So let me now define the derivative
2
00:00:05,105 --> 00:00:08,082
a little bit more formally.
3
00:00:08,082 --> 00:00:09,848
Returning to the example we started with,
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00:00:09,848 --> 00:00:13,097
average speed, from time 1 to time 2
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00:00:13,097 --> 00:00:15,910
is the distance travelled, during that time interval,
6
00:00:15,910 --> 00:00:18,325
divided by the time interval itself,
7
00:00:18,325 --> 00:00:21,617
and then we can say, well, let t2,
8
00:00:21,617 --> 00:00:25,139
the second time, get closer and closer to t1,
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00:00:25,139 --> 00:00:27,919
that's equivalent to saying
10
00:00:27,919 --> 00:00:32,004
if I think about average speed as this:
11
00:00:32,004 --> 00:00:34,831
change in position, over a change in time,
12
00:00:34,831 --> 00:00:45,195
let this time interval, delta t, get closer and closer to 0,
13
00:00:45,195 --> 00:00:47,247
so, if I'm thinking in terms of this notation:
14
00:00:47,247 --> 00:00:48,743
delta s over delta t,
15
00:00:48,743 --> 00:00:52,184
I let delta t get closer and closer and closer to 0
16
00:00:52,184 --> 00:00:55,782
I can't just plug in 0 because that'll be dividing by 0,
17
00:00:55,782 --> 00:00:58,778
but I let it get closer and closer and closer to 0
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00:00:58,778 --> 00:01:01,151
and I monitor what this number becomes,
19
00:01:01,151 --> 00:01:02,411
what this ratio becomes,
20
00:01:02,411 --> 00:01:05,246
and the result is the instantaneous speed,
21
00:01:05,246 --> 00:01:09,159
and the notation for that often is: ds over dt,
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00:01:09,159 --> 00:01:15,152
so instead of these triangles - delta's, which means big change,
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00:01:15,152 --> 00:01:22,501
we use these little d's - which mean: instantaneous change.
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00:01:29,099 --> 00:01:32,180
So, finally, and a little bit more generally,
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00:01:32,180 --> 00:01:33,885
let's think about the derivative of x
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00:01:33,885 --> 00:01:37,437
so here x could be any function that changes at all.
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00:01:37,437 --> 00:01:40,992
this could be the position, it could be temperature, it could be population,
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00:01:40,992 --> 00:01:43,330
it's just some generic function that changes,
29
00:01:43,330 --> 00:01:45,073
it doesn't have to be position,
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00:01:45,073 --> 00:01:48,743
and we could ask: How does it change instantaneously?
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00:01:48,743 --> 00:01:52,102
So, the derivative of x, dx/dt,
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00:01:52,102 --> 00:01:54,837
is the instantaneous rate of change at x
33
00:01:54,837 --> 00:01:57,805
that's the key thing you need to know from this subunit:
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00:01:57,820 --> 00:02:01,711
the derivative is the instantaneous rate of change.
35
00:02:01,711 --> 00:02:07,320
Let me write just a few other ways to picture this -
36
00:02:07,320 --> 00:02:11,178
it's also the slope of x of t. x of t could be a curve -
37
00:02:11,178 --> 00:02:14,822
but that's okay - we saw on the previous video
38
00:02:14,822 --> 00:02:21,145
how to think about slopes have curves,
39
00:02:24,069 --> 00:02:26,139
and, for those of you who have had calculus
40
00:02:26,139 --> 00:02:27,263
here's the formal definition:
41
00:02:27,263 --> 00:02:29,306
it's the limit as h goes to 0
42
00:02:29,306 --> 00:02:33,097
x of t plus h, minus x of t over h,
43
00:02:33,097 --> 00:02:35,386
which I just like to think of as:
44
00:02:35,386 --> 00:02:40,815
change in x, over change in t,
45
00:02:40,815 --> 00:02:44,323
in the limit that these changes become very, very small.
46
00:02:44,323 --> 00:02:45,990
So, again, the derivative is:
47
00:02:45,990 --> 00:02:49,239
the instantaneous rate of change of x
48
00:02:49,239 --> 00:02:51,617
It's telling you how fast x is changing
49
00:02:51,617 --> 00:02:55,740
and, you might get different values, depending on the time
50
00:02:55,740 --> 00:03:00,784
so this allows for a function x that's growing
51
00:03:00,784 --> 00:03:03,401
at different rates, at different times.
52
00:03:03,401 --> 00:03:06,487
Lastly, just a little bit notation:
53
00:03:06,487 --> 00:03:09,538
so this idea of a derivative
54
00:03:09,538 --> 00:03:11,407
is quite often
55
00:03:11,407 --> 00:03:16,217
to note it this way: which should be read x-prime,
56
00:03:16,217 --> 00:03:20,075
and in physics and engineering,
57
00:03:20,075 --> 00:03:23,808
one often uses this notation: x-dot,
58
00:03:23,808 --> 00:03:26,929
which the stands for time derivative.
59
00:03:26,929 --> 00:03:29,856
In this course are almost always use this first derivative.
60
00:03:29,856 --> 00:03:32,983
I might occasionally, maybe by mistake, use this one,
61
00:03:32,983 --> 00:03:36,299
I won't use this one at all, but you're likely to see this
62
00:03:36,299 --> 00:03:41,382
in other texts or papers particularly if it's from
63
00:03:41,382 --> 00:03:45,382
the physics or engineering community.