Higher order derivatives | Chapter 10, Essence of calculus

3Blue1Brown · May 12, 2026

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Transcript ~873 words · 5:38

0:04

In the next chapter about Taylor series, I make

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frequent reference to higher order derivatives.

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And if you're already comfortable with second derivatives,

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third derivatives, and so on, great!

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Feel free to just skip ahead to the main event now.

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You won't hurt my feelings.

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But somehow, I've managed not to bring up higher

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order derivatives at all so far in this series.

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So for the sake of completeness, I thought I'd give you

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this little footnote just to go over them very quickly.

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I'll focus mainly on the second derivative, showing what it looks like in the context

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of graphs and motion, and leave you to think about the analogies for higher orders.

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Given some function f of x, the derivative can be

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interpreted as the slope of this graph above some point, right?

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A steep slope means a high value for the derivative,

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a downward slope means a negative derivative.

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So the second derivative, whose notation I'll explain in just a moment,

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is the derivative of the derivative, meaning it tells you how that slope is changing.

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The way to see that at a glance is to think about how the graph of f of x curves.

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At points where it curves upwards, the slope is increasing,

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and that means the second derivative is positive.

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At points where it's curving downwards, the slope is decreasing,

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so the second derivative is negative.

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For example, a graph like this one has a very positive second derivative at the point 4,

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since the slope is rapidly increasing around that point,

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whereas a graph like this one still has a positive second derivative at the same point,

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but it's smaller, the slope only increases slowly.

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At points where there's not really any curvature, the second derivative is just 0.

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As far as notation goes, you could try writing it like this,

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indicating some small change to the derivative function,

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divided by some small change to x, where as always the use of this letter d

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suggests that what you really want to consider is what this ratio approaches as dx,

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both dx's in this case, approach 0.

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That's pretty awkward and clunky, so the standard is

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to abbreviate this as d squared f divided by dx squared.

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And even though it's not terribly important for getting an intuition for the second

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derivative, I think it might be worth showing you how you can read this notation.

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To start off, think of some input to your function,

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and then take two small steps to the right, each one with a size of dx.

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I'm choosing rather big steps here so we'll be able to see what's going on,

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but in principle keep in the back of your mind that dx should be rather tiny.

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The first step causes some change to the function, which I'll call df1,

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and the second step causes some similar but possibly slightly different change,

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which I'll call df2.

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The difference between these changes, the change in how the function changes,

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is what we'll call ddf.

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You should think of this as really small, typically proportional to the size of dx2.

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So if, for example, you substituted in 0.01 for dx,

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you would expect this ddf to be about proportional to 0.0001.

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The second derivative is the size of this change to the change,

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divided by the size of dx2, or more precisely,

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whatever that ratio approaches as dx approaches 0.

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eleration. Given some movement along a line, suppose you have some function that

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records the distance traveled versus time, maybe its graph looks like this,

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steadily increasing over time. Then its derivative tells you velocity at each

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point in time, for example the graph might look like this bump,

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increasing up to some maximum, and decreasing back to zero.

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So the second derivative tells you the rate of

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Maybe the most visceral understanding of the second

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derivative is that it represents acceleration.

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Given some movement along a line, suppose you have some function

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that records the distance traveled versus time,

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maybe its graph looks something like this, steadily increasing over time.

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Then its derivative tells you velocity at each point in time,

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for example the graph might look like this bump, increasing up to some maximum,

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and decreasing back to zero.

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The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero,

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it means that the strength of the acceleration itself is changing.

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One of the most useful things about higher order derivatives is how they help us in

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approximating functions,

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In this example, the second derivative is positive for the first half of the journey,

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which indicates speeding up, that's the sensation of being pushed back into

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your car seat, or rather, having the car seat push you forward.

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A negative second derivative indicates slowing down, negative acceleration.

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The third derivative, and this is not a joke, is called jerk.

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So if the jerk is not zero, it means the strength of the acceleration itself is changing.

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One of the most useful things about higher order derivatives is

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how they help us in approximating functions, which is exactly the

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topic of the next chapter on Taylor series, so I'll see you there.

— end of transcript —

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