Higher order derivatives | Chapter 10, Essence of calculus

A very quick primer on the second derivative, third derivative, etc.

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Italian: hi-anji
Vietnamese: ngvutuan2811

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Various social media stuffs:

[derekdziobek]

The change of position over time is velocity.
The change of velocity over time is acceleration.
The change of acceleration over time is a jerk.
The change of a jerk over time is an election.

[RandomDays]

The 4th, 5th, and 6th derivatives are Snap, Crackle, and Pop, respectively.

[pillsofpink]

My calculus professor is sending us links to these vids instead of having a zoom lecture. So congrats on teaching MATH155 at Colorado State University.

[tobybartels]

4:48 : I have to correct this, because it confuses my students too. You said ‘A negative second derivative [of displacement] indicates slowing down’, but that's only correct _if_ the velocity is positive. As you noted in the video on derivatives, a negative velocity means that you are headed in the negative direction. And in that case, a negative acceleration means that you are _speeding up, _ with the velocity becoming even more negative, while a _positive_ acceleration means that you are slowing down. If you want a quantity that's positive when you're speeding up and negative when you're slowing down, then you need to take the derivative of the _speed, _ that is of the absolute value of the velocity, so the second derivative of the total distance travelled, but _not_ the second derivative of the displacement. (Arguably, this fits more with the way we use the word ‘acceleration’ in ordinary language, but the technical meaning is the second derivative of displacement.)

As an aside, this disparity becomes even more extreme if you're moving in multiple dimensions of space. In that case, the displacement, velocity, and acceleration are all vectors, and it doesn't make sense to say that they are positive or negative as such. Then the speed is the magnitude of the velocity vector, and the derivative of the speed is again positive if you're speeding up and negative if you're slowing down. But now it's also possible for the derivative of the speed to be zero, even if the acceleration is nonzero! In that case, the speed is constant but the velocity is not, because you're changing direction.

[dannyundos]

I think Korean is funnier here. After "velocity", you just add "가".
Displacement = 변위
Velocity = 속도
Acceleration = 가속도
Jerk = 가가속도
4th derivative = 가가가속도
5th derivative = 가가가가속도
6th derivative = 가가가가가속도
...
nth derivative = (가)^(n-1)속도

[xtuner]

Who dislikes this video is a 3rd derivative

[mesplin]

3:47
"Interestingly, there is a notion in math called the 'exterior derivative' which treats this 'd' as having a more independent meaning, though it's less relatable to the intuitions I've introduced in this series"

[idrisShiningTimes]

Beautiful explanation, visualisation, and most importantly, the simplicity you always use to explain complex terms. Love it

[SuperElephant]

-5 >> Absounce
-4 >> Abserk
-3 >> Abseleration
-2 >>Absity
-1 >>Absement
0 >> Displacement
1 >> Velocity
2 >> Acceleration
3 >> Jerk
4 >> Jounce

I really had a hard time understanding Less than 0 and more than 2...
Can anyone make a video to explain it all??

[aajjeee]

Position
Velocity
Acceleration
Jerk
snap
Crackle
Pop

[patrickhodson]

Nowadays everyone is releasing non-episodes in the same universe. First there was _Rogue One: a Star Wars Story, _ and now we've got _Higher Order Derivatives: a Calculus Story._

[hahahasan]

You should definitely do a video on the gamma function and fractional derivatives.

[SandeepSingh-qrdk]

Hello Grant, I really admire your videos as you can see I am watching these again even after two years. Please do a series of animations on Complex Analysis and Transforms (laplace, Fourier and Z).

[ryanlira]

can u do an essence of differential equations? ubhave no idea how much i love these

[chaosui]

3:31 much clear now: the second derivative is treated as the difference of two first derivative: if its positive, it increases

[kjekelle]

0:00 intro
0:39 derivative of the derivative
1:53 notation
3:58 intuition
5:05 outro

[unclegranpawafiaahmedyahia]

Ces vidéos sont supers..je conseil ;
grand merci 3bleus 1marron..

[loganstrong]

I took Calculus (1 2 and 3) back in high school. I am watching this series for probably the third time because these were all the same intuitions I had that helped me understand the subject the first time around. Keep up the great work with all your videos!

[marcinukaszyk]

I just want to say:thank you! I learned a lot

[existentialchaos]

When I was around 9, I realized that all number patterns have "layers" underneath them. The first layer below it would be how much it increased by each time, the 2nd would be how much the 1st layer increased by each time, and so on. I had this theory that every pattern, if you "peel" the layers enough, it would always reach a layer where all terms would be the same number, and that was the "base layer" that every pattern was made out of (now I know this is true for polynomials functions), and each pattern could be classified by the number of layers it had. For example, for a pattern like 1, 4, 9, 16, etc., it would be a 3rd layer pattern because the layer underneath, or the 2nd layer, is 3, 5, 7, 9, ..., and the layer underneath that, or the 1st layer, is just 2, 2, 2, ...
I realized I just basically found out the concepts of arithmetic sequences, polynomial degrees, derivatives, and possible Taylor Series.