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.