Nth derivative formula for high degree power rule

Question

I was wondering if there is a general formula for finding some $n$-th derivative. I came up with this (the Google doc has the math with proper formatting). Can someone tell me if this is correct and if it has been done before (it probably has)? If it has been done before, can someone please link the source?

Also, is there any way to make it more general?

Thanks.

Answer 1

There are formulas for $n$-th derivate for specific functions where you can find a pattern. There is no special source to link since it is something super general. Everyone does it with several functions.

I will let you here some of them.

EXAMPLE $1$:

$$f(x)=x\cdot e^{-x}$$ $$f^{n)}(x)=((-1)^n)(x\cdot e^{-x})+((-1)^{n+1}\cdot n)(e^{-x})$$

EXAMPLE $2$:

$$g(x)=\frac{1}{1-x}$$ $$g^{n)}(x)=\frac{n!}{(1-x)^{n+1}}$$

Yours is one more example. It is correct. You have found a pattern and you have defined the $n$-th derivate for your function.