Question

How do you find the nth derivative of the function `f(x)=1/x`?

Answer 1

`f^((n)) = (-1)^n n! x^(-(n+1))`, or `f^((n)) = ((-1)^n n!)/x^(n+1)`

Explanation:

`f(x) = 1/x = x^-1`

When `{ (n=1,=>f^((n))=f'(x),=-1x^(-2)), (n=2,=>f^((n))=f''(x),=+1*2x^(-3)), (n=3, =>f^((n))=f'''(x),=-1*2*3x^(-4)), (n=4, =>f^((n))=f''''(x),=+1*2*3*4x^(-5)), (,vdots,vdots) :}`

We can see a clear pattern forming and we can conclude that in the general case we have:

`f^((n)) = (-1)^n n! x^(-(n+1))`, or `f^((n)) = ((-1)^n n!)/x^(n+1)`