Interesting Successive Differentiation Formula

Hello, I would like to share this video on Faa di Bruno's formula, a formula that generalizes the chain rule to higher-order derivatives. What's so interesting about it is the connections to integer partitions and discrete math. Hope you like it:

As a Civil Engineer having taken math courses through differential equations. I don't understand the practical use for this.

Please explain.

There may not be any practical use for it really, except as an object of fascination for mathematicians.

Sometimes we can just get curious and start inspecting: what happens if we keep taking the derivative over and over and over again, in the chain rule?  It is only the kind of thing you would probably do if you were bored, or if you just have some kind of fantastic math imagination like Ramanujan or something.  Sometimes a person just starts down a path and gets obsessed with that particular path and that is how such formulas come to be.  And it might happen to different people in different places and times so mathematicians enjoy writing this down so that others can share in the appreciation.  It's like asking what is the purpose of a piece of music (I'm thinking like Bach music, https://www.youtube.com/watch?v=rGgG-0lOJjk, not some kind of pop tune on the radio like the one about nose and toes https://www.youtube.com/watch?v=AWGqoCNbsvM)

This is a really beautiful post, thanks for this. I agree, I can't think of any practical uses for Faa di Bruno's formula either, but as with many things in pure math, there could be practical uses that we find out about in the future. I see the formula more as an interesting connection between something discrete like integer partitions and something continuous like differentiation.  

Agreeing?! I thought we were supposed to always argue everything; thank you for agreeing.  I may have to turn around and argue the opposite point of view, that there can be some practical value to that formula.

Here are more thoughts about the video:

One question a math or engineering student might want to consider: of what use are high order derivatives such as the 5th through 10th order derivatives?  In my mind I have always held them to be doubtful and dubious.  The fourth order derivative: I believe I've been told by my high school calc teacher long ago that humans can see if there is a discontinuity in the 4th order derivative of a shiny surface (such as a Corvette or a Porsche, etc.  One calc student I had, her surname was Ferrera and I did not know it was a model of Porsche.)

Having now watched the video, I think the video could be improved in several ways small and large.

Quibbles or Questions:

1) It is called the order of the derivative, not the 'derivative exponent'.  If exponents are not about exponentiation, as in the inverse of the natural logarithm,  or powers of ten, then they should consider not being called exponents.  Actually it is a pretty big question since I am not sure if an exponent denotes a position of a smaller symbol a little higher than a bigger symbol on the page - I hope there is some other word for it.

2) he has some equations with pluses in between them; i wish that there would be parentheses around the equations, so that reading it in the wrong order can not cause a person to see a false statement.  I am way too pedantic about this. But I dislike an algebra system where we have to constantly reconsider the order of operations, after having learned them solidly in earlier courses.  In Botswana my students would look at 3+7+4 and write 3+7=10+4=14 which looks simply false to me since it says 3+7 = 14. We did not understand each other very well.  I should have wasted less red ink on it.

3) Purposes of the formula might be clearer in the beginning of the video if the video could show the formula briefly for the 2nd and 3rd and 4th order derivatives, and maybe even work them out, so the mind can more quickly grasp how the generalization of the formula might proceed, before it is displayed in its full ugly glory, and start to glimpse how partitions might play a factor.  There's some salesmanship going on about this formula.  Actually I'm not sure if that's a good teaching method or not.  The scary looking formula might put people off to start with, where looking at the derivatives for patterns might be more interesting for people to see.  It's a question of presentation.