Repeated differentiation of $y \cos(x)$

Question

When solving a problem to compute the Taylor expansion of $\tan(x)$ (hence need to differentiate the function many times), I tried the following:

If $y = \tan x$, then $$ y \cos x = \sin x $$

Since the differentiation of the right side is trivial, focus on the left side. Differentiating the left side,

$$ y' \cos x - y \sin x $$

Differentiating further:

$$ y'' \cos x - 2 y' \sin x - y \cos x $$

$$ y''' \cos x -3 y'' \sin x - 3 y' \cos x + y \sin x $$

$$ y'''' \cos x -4 y''' \sin x - 6 y'' \cos x + 4 y' \sin x + \cos x $$

and so on.

The results resemble the binomial expansion; powers area replaced by differentiation, and each term is multiplied by $\cos x, -\sin x, -\cos x, \sin x, ...$ from left to right. I find this to be useful since I can recover the higher order derivatives relatively easily, with less chance of errors.

My question is if this outcome is somehow related to other math areas. Does it have any name? Or does it have any connection to any advanced math areas, possibly, but not limited to, real/complex analysis? Any suggestions are appreciated.

Answer 1

What you noticed is known as General Leibniz Rule which states that

$$(fg)^{(n)}(x)~=~\sum_{k=0}^n \binom nk f^{(n-k)}(x)g^{(k)}(x)\tag1$$

Formula $(1)$ can be specified for your case by setting $f(x)=y$ and $g(x)=\cos(x)$ respectively.