Question

Consider the function f denoted by:


`f(x) = ln(x)`
Find the nth derivative of f(x) denoted by:

`f {}^((n)) (x )`
Irrelevant answers will be reported immediately.
​

Answer 1

Explanation:

Let take the first derivative

`(d)/(dx) ln(x)) = x {}^( - 1)`

The second derivative

`- {x}^( - 2)`

The third derivative

`2 {x}^( - 3)`

The fourth derivative

`- 6 {x}^( - 4)`

The fifth derivative

`24 {x}^( - 5)`

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

`{x}^( - n)`

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

`(n - 1)`

So far, our nth derivative is

`(n - 1)!x {}^( - n)`

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)

`- 1 {}^(n -1) (n - 1) ! {x}^(-n)`

That is our nth derivative