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
such, great!
Feel free to skip right 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 this
series, so for the sake of completeness, I thought I’d give this little footnote to
very briefly go over them.
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(x), the derivative can be interpreted as the slope of its graph above
some input, right?
A steep slope means a high value for the derivative, a downward slope means a negative derivative.
The second derivative, whose notation I’ll explain in a moment, is the derivative of
the derivative, meaning it tells you how that slope is changing.
The way to see this at a glance is to think of how the graph of f(x) curves.
At points where it curves upward, the slope is increasing, so the second derivative is
positive.
At points where it curves downward, the slope is decreasing, so the second derivative is
negative.
For example, a graph like this has a very positive second derivative at the input 4,
since the slope is rapidly increasing around that point, whereas a graph like this still
has a positive second derivative at that same point, but it’s smaller, since the slope
is increasing only slowly.
At points where there’s not really any curvature, the second derivative is 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 that letter d suggests that you really want to consider what this ratio approach
as dx, both dx’s in this case, approach 0.
That’s pretty awkward and clunky, so the standard is to abbreviate it as d2f/dx2.
It’s not terribly important for getting an intuition of the second derivative, but
perhaps it’s worth showing how you can read this notation.
Think of starting at some input to your function, and taking two small steps to the right, each
with a size dx.
I’m choosing rather big steps here so that we’ll better see what’s going on, but
in principle think of them as rather tiny.
The first step causes some change to the function, which I’ll call df1, and the second step
causes some similar, but possibly slightly different change, which I’ll call df2.
The difference between these; the change in how the function changes, is what we’ll
call d(df).
You should think of this as really small, typically proportional to the size of (dx)2.
So if your choice for dx was 0.01, you’d expect this d(df) to be proportional to 0.001.
And the second derivative is the size of this change to the change, divide by the size of
(dx)2.
Or, more precisely, it’s whatever that ratio approaches as dx approaches 0.
Even though it’s not like the letter d is a variable being multiplied by f, for the
sake of more compact notation you write this as d2f/dx2, and you don’t bother with any
parentheses on the bottom.
Maybe the most visceral understanding of the second derivative is that it represents acceleration.
Given some movement along a line, suppose you have some function that records distance
traveled vs. time, and maybe its graph looks something like this, steadily increasing over
time.
Then its derivative tells you velocity at each point in time, right?
For the example, the graph might look like this bump, increasing to some maximum, then
decreasing back to 0.
So its second derivative tells you the rate of change for velocity, the acceleration at
each point in time.
In the example, the second derivative is positive for the first half of the journey, which indicates
indicates speeding up.
That’s sensation of being pushed back into your car seat with a constant force.
Or rather, having the car seat push you with a constant force.
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 the strength of the acceleration itself is changing.
One of the most useful things about higher order derivatives is how they help in approximating
functions, which is the topic of the next chapter on Taylor series, so I’ll see you
there.