Question

How do I find a general formula for `f^(n)(x)` if `f(x)=x^-4`?

Answer 1

`x^{(-1)^n 4^n}`

Explanation:

`f(x) = x^-4`, so
`f^2(x) = (x^-4)^-4=x^16,quad f^3(x) = x^-64`

From this we may conjecture

`f^n(x)= x^{(-1)^n 4^n}`

Let the conjecture be true for a particular positive integer `n`. Then
`f^{n+1}(x) = f(f^n(x)) = (x^{(-1)^n 4^n})^-4 = x^{(-1)^{n+1} 4^{n+1)}`
so the formula is valid for `n+1` if it is valid for `n`. Since the formula is valid for `n=1`, it is valid for all positive integers according to the principle of mathematical induction.