Explaining how to take derivatives in this form: $\frac{n!}{(n-1)!}x^{\frac{(n-1)!}{(n-2)!}}$

Question

I was going through some calculus integration and had the idea of implementing derivates while in this form: $\frac{n!}{(n-1)!}x^{\frac{(n-1)!}{(n-2)!}}$

It surprisingly seems to work:

from sympy import *
from numpy import *
x = symbols('x')
def derivative(n):
    derive = (factorial(n)/factorial(n-1))*x**(factorial(n-1)/factorial(n-2))
    return derive

derivative(3)

$=3x^2$

Although I wished to know as to whether there was a stronger understanding beyond this approach, and perhaps it's used elsewhere or has a specific name for it which I have not come across? Furthermore, what's the theoretical understanding for this approach working?

Answer 1

We simply have that by definition

$$n!=n\cdot (n-1)\cdot (n-2)\cdots \cdot 2\cdot 1$$

therefore

$$\frac{n!}{(n-1)!} =n$$

$${\frac{(n-1)!}{(n-2)!}} =n-1$$

and then

$$\frac{n!}{(n-1)!}x^{\frac{(n-1)!}{(n-2)!}} =nx^{n-1}$$