I Why Does This Higher Order Derivative Equation Hold?

[LauwranceGilbert]

mod: moved from homework

Does anyone know why and when this equation holds? I have searched online but cannot find the reason or the rules for the higher order derivatives.

 

[jedishrfu]

Can you provide some context of where you found this equation and what you were investigating?

It looks like ordinary power rules applied to derivatives ie derivative chain rule.

https://en.wikipedia.org/wiki/Chain_rule

There's a section further into the article talking about generalizations of the rule:

Higher derivatives[edit]
Faà di Bruno's formula generalizes the chain rule to higher derivatives. Assuming that y = f(u) and u = g(x), then the first few derivatives are:

\begin{aligned}{\frac {dy}{dx}}&={\frac {dy}{du}}{\frac {du}{dx}}\\[4pt]{\frac {d^{2}y}{dx^{2}}}&={\frac {d^{2}y}{du^{2}}}\left({\frac {du}{dx}}\right)^{2}+{\frac {dy}{du}}{\frac {d^{2}u}{dx^{2}}}\\[4pt]{\frac {d^{3}y}{dx^{3}}}&={\frac {d^{3}y}{du^{3}}}\left({\frac {du}{dx}}\right)^{3}+3\,{\frac {d^{2}y}{du^{2}}}{\frac {du}{dx}}{\frac {d^{2}u}{dx^{2}}}+{\frac {dy}{du}}{\frac {d^{3}u}{dx^{3}}}\\[4pt]{\frac {d^{4}y}{dx^{4}}}&={\frac {d^{4}y}{du^{4}}}\left({\frac {du}{dx}}\right)^{4}+6\,{\frac {d^{3}y}{du^{3}}}\left({\frac {du}{dx}}\right)^{2}{\frac {d^{2}u}{dx^{2}}}+{\frac {d^{2}y}{du^{2}}}\left(4\,{\frac {du}{dx}}{\frac {d^{3}u}{dx^{3}}}+3\,\left({\frac {d^{2}u}{dx^{2}}}\right)^{2}\right)+{\frac {dy}{du}}{\frac {d^{4}u}{dx^{4}}}.\end{aligned}

Here's a presentation where they use something like this for trig derivatives:

https://www.cs.drexel.edu/classes/Calculus/MATH121_Fall02/lecture14.pdf

 

[Ray Vickson]

mod: moved from homework

Does anyone know why and when this equation holds? I have searched online but cannot find the reason or the rules for the higher order derivatives.

View attachment 213801

The first equality is easy: the $(4M+4)$th derivative of $F$ is just the 4th derivative of the $(4M)$th derivative. The second equality is false, in general, because there are many counterexamples. The $(4M+4)$th derivative is usually not a constant ($= (-4)^M$ ) times the 4th derivative.