Step-by-step explanation:
Let take the first derivative
[tex] \frac{d}{dx} ln(x)) = x {}^{ - 1} [/tex]
The second derivative
[tex] - {x}^{ - 2} [/tex]
The third derivative
[tex]2 {x}^{ - 3} [/tex]
The fourth derivative
[tex] - 6 {x}^{ - 4} [/tex]
The fifth derivative
[tex]24 {x}^{ - 5} [/tex]
Let create a pattern,
The values always have x in it so
our nth derivative will have x in it.
The nth derivative matches the negative nth power so the nth derivative so far is
[tex] {x}^{ - n} [/tex]
Next, lok at the constants. They follow a pattern of 1,2,6,24,120). This is a factorial pattern because
1!=1
2!=2
3!=6
4!=24
5!=120 and so on. Notice how the nth derivative has the constant of the factorial of the precessor
so our constant are
[tex](n - 1)[/tex]
So far, our nth derivative is
[tex](n - 1)!x {}^{ - n} [/tex]
Finally, notice for the odd derivatives we are Positve and for the even ones, we are negative, this means we are raised -1^(n-1)
[tex] - 1 {}^{n -1} (n - 1) ! {x}^{-n} [/tex]
That is our nth derivative
