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

1 Answer
Feb 20, 2018

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.