Higher order derivatives | Chapter 10, Essence of calculus

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, a downward slope means a 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, 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 4, since the slope is rapidly increasing around that point, whereas a graph like this one still has a positive second derivative at the same point, but it's smaller, the slope only increases slowly. At points where there's not really any curvature, the second derivative is just 0. As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by x, where as always the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx, both dx's in this case, approach 0. 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 off, think of some input to your function, and then take two small steps to the right, each one with a size of dx. I'm choosing rather big steps here so 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 df1, and the second step causes some similar but possibly slightly different change, which I'll call df2. 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 dx2. So if, for example, you substituted in 0. 01 for dx, you would expect this ddf to be about proportional to 0. 0001.

The second derivative is the size of this change to the change, divided by the size of dx2, or more precisely, whatever that ratio approaches as dx approaches 0. eleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. So the second derivative tells you the rate of 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 the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time. 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, 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 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.