In the next chapter about Taylor series, I make frequent reference to higher-order
derivatives. And if you’re already comfortable with second derivatives, third derivatives, and so
on, great! Feel free to just skip ahead to the main event now. You won’t hurt my feelings. But somehow, I’ve managed not to bring up higher-order derivatives at all so far in
this series. So for the sake of completeness, I thought I’d give you this little footnote, just to
go over them very quickly. I’ll focus mainly on the second derivative, showing what it looks like in the context
of graphs and motion. And leave you to think about the analogies for higher orders.
Given some function 𝑓 of 𝑥, the derivative can be interpreted as the slope of this
graph above some point, right? A steep slope means a high value for the derivative. Downward slope means negative derivative. So the second derivative, whose notation I’ll explain in just a moment, is the
derivative of the derivative. Meaning, it tells you how that slope is changing. The way to see that at a glance is to think about how the graph of 𝑓 of 𝑥
curves. At points where it curves upwards like this, the slope is increasing. And that means the second derivative is positive. At points where it’s curving downwards, the slope is decreasing. So the second derivative is negative.
For example, a graph like this one has a very positive second derivative at the point
four. Since the slope is rapidly increasing around that point. Whereas a graph like this one still has a positive second derivative at that same
point, but it’s smaller. I mean, the slope only increases slowly. At points where there’s not really any curvature, the second derivative is just
zero. As far as notation goes, you could try writing it like this, indicating some small
change to the derivative function divided by some small change to 𝑥. Where, as always, the use of this letter d suggests that what you really wanna
consider is what this ratio approaches as d𝑥, both d𝑥s in this case, approach
That’s pretty awkward and clunky. So the standard is to abbreviate this as d squared 𝑓 divided by d𝑥 squared. And even though it’s not terribly important for getting an intuition for the second
derivative. I think it might be worth showing you how you can read this notation. To start of, think of some input to your function and then take two small steps to
the right, each one with a size of d𝑥. And choosing rather big steps here so that we’ll be able to see what’s going on. But in principle, keep in the back of your mind that d𝑥 should be rather tiny. The first step causes some change to the function, which I’ll call d𝑓 one. And the second step causes some similar but possibly slightly different change, which
I’ll call d𝑓 two. The difference between these changes, the change in how the function changes, is what
we’ll call dd𝑓.
You should think of this as really small, typically proportional to the size of d𝑥
squared. So if, for example, you substituted in 0.01 for d𝑥, you would expect this dd𝑓 to be
about proportional to 0.0001. And the second derivative is the size of this change to the change divided by the
size of d𝑥 squared. Or, more precisely, it’s whatever that ratio approaches as d𝑥 approaches zero. Even though, it’s not like this letter d is a variable being multiplied by 𝑓. For the sake of more compact notation, you’d write it as d squared 𝑓 divided by d𝑥
squared. And you don’t typically bother with any parentheses on the bottom. Maybe the most visceral understanding of the second derivative is that it represents
Given some movement along a line, suppose you have some function that records the
distance traveled versus time. Maybe its graph looks something like this, steadily increasing over time. Then, its derivative tells you velocity at each point in time, right? For example, the graph might look like this bump, increasing up to some maximum and
then decreasing back to zero. So the second derivative tells you the rate of change for the velocity, which is the
acceleration at each point in time.
In this example, the second derivative is positive for the first half of the journey,
which indicates speeding up. That’s the sensation of being pushed back into your car seat or, rather, having the
car seat push you forward. A negative second derivative indicates slowing down, negative acceleration. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is
changing. One of the most useful things about higher order derivatives is how they help us in
approximating functions. Which is exactly the topic of the next chapter on Taylor series. So I’ll see you there.