[00:04] In the next chapter about Taylor series, I make
[00:06] frequent reference to higher order derivatives.
[00:10] And if you're already comfortable with second derivatives,
[00:12] third derivatives, and so on, great!
[00:14] Feel free to just skip ahead to the main event now.
[00:16] You won't hurt my feelings.
[00:18] But somehow, I've managed not to bring up higher
[00:21] order derivatives at all so far in this series.
[00:24] So for the sake of completeness, I thought I'd give you
[00:26] this little footnote just to go over them very quickly.
[00:29] I'll focus mainly on the second derivative, showing what it looks like in the context
[00:34] of graphs and motion, and leave you to think about the analogies for higher orders.
[00:40] Given some function f of x, the derivative can be
[00:43] interpreted as the slope of this graph above some point, right?
[00:47] A steep slope means a high value for the derivative,
[00:50] a downward slope means a negative derivative.
[00:53] So the second derivative, whose notation I'll explain in just a moment,
[00:57] is the derivative of the derivative, meaning it tells you how that slope is changing.
[01:03] The way to see that at a glance is to think about how the graph of f of x curves.
[01:08] At points where it curves upwards, the slope is increasing,
[01:12] and that means the second derivative is positive.
[01:17] At points where it's curving downwards, the slope is decreasing,
[01:21] so the second derivative is negative.
[01:26] For example, a graph like this one has a very positive second derivative at the point 4,
[01:32] since the slope is rapidly increasing around that point,
[01:36] whereas a graph like this one still has a positive second derivative at the same point,
[01:42] but it's smaller, the slope only increases slowly.
[01:46] At points where there's not really any curvature, the second derivative is just 0.
[01:53] As far as notation goes, you could try writing it like this,
[01:57] indicating some small change to the derivative function,
[02:01] divided by some small change to x, where as always the use of this letter d
[02:06] suggests that what you really want to consider is what this ratio approaches as dx,
[02:12] both dx's in this case, approach 0.
[02:15] That's pretty awkward and clunky, so the standard is
[02:19] to abbreviate this as d squared f divided by dx squared.
[02:24] And even though it's not terribly important for getting an intuition for the second
[02:28] derivative, I think it might be worth showing you how you can read this notation.
[02:33] To start off, think of some input to your function,
[02:36] and then take two small steps to the right, each one with a size of dx.
[02:42] I'm choosing rather big steps here so we'll be able to see what's going on,
[02:45] but in principle keep in the back of your mind that dx should be rather tiny.
[02:50] The first step causes some change to the function, which I'll call df1,
[02:55] and the second step causes some similar but possibly slightly different change,
[03:01] which I'll call df2.
[03:03] The difference between these changes, the change in how the function changes,
[03:08] is what we'll call ddf.
[03:12] You should think of this as really small, typically proportional to the size of dx2.
[03:18] So if, for example, you substituted in 0.01 for dx,
[03:23] you would expect this ddf to be about proportional to 0.0001.
[03:29] The second derivative is the size of this change to the change,
[03:34] divided by the size of dx2, or more precisely,
[03:37] whatever that ratio approaches as dx approaches 0.
[03:43] eleration. Given some movement along a line, suppose you have some function that
[03:45] records the distance traveled versus time, maybe its graph looks like this,
[03:48] steadily increasing over time. Then its derivative tells you velocity at each
[03:51] point in time, for example the graph might look like this bump,
[03:53] increasing up to some maximum, and decreasing back to zero.
[03:56] So the second derivative tells you the rate of
[03:59] Maybe the most visceral understanding of the second
[04:01] derivative is that it represents acceleration.
[04:05] Given some movement along a line, suppose you have some function
[04:08] that records the distance traveled versus time,
[04:11] maybe its graph looks something like this, steadily increasing over time.
[04:16] Then its derivative tells you velocity at each point in time,
[04:20] for example the graph might look like this bump, increasing up to some maximum,
[04:24] and decreasing back to zero.
[04:27] The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero,
[04:29] it means that the strength of the acceleration itself is changing.
[04:31] One of the most useful things about higher order derivatives is how they help us in
[04:33] approximating functions,
[04:34] In this example, the second derivative is positive for the first half of the journey,
[04:39] which indicates speeding up, that's the sensation of being pushed back into
[04:43] your car seat, or rather, having the car seat push you forward.
[04:47] A negative second derivative indicates slowing down, negative acceleration.
[04:54] The third derivative, and this is not a joke, is called jerk.
[04:57] So if the jerk is not zero, it means the strength of the acceleration itself is changing.
[05:06] One of the most useful things about higher order derivatives is
[05:09] how they help us in approximating functions, which is exactly the
[05:13] topic of the next chapter on Taylor series, so I'll see you there.