FInd the 1000th derivative of Ln(x+1)

Question

I am struggling as i know the sequence but got no idea how to describe it mathematically!

Answer 1

Although I did not prove it by mathematical induction it looks as if the nth derivative is (-1)ⁿ⁺¹(n-1)! (x+1)^{-n.  If you agree, substitute n=1000.  Good luck!}

Answer 2

1st derivative:  d ln(x+1)/dx = (x+1)^-1

2nd derivative:  d(x+1)^-1/dx = -(x+1)^-2

3rd derivative:  2(x+1)^-3

4th derivative: -(3*2)(x+1)^-4

...

nth derivative: (-1)^n-1(n-1)!(x+1)^-n

 

Plug in n = 1000