Prove that $f^{n}=(-1^{n})(\frac{1}{x^{n+1}})$ using the definition of the derivative

Question

I know how to prove that $y´=-\frac{1}{x^2}$ using the definition of the derivative, but how do you prove it for all the derivatives? Thanks so much!

Answer 1

let $y(x) = x^{-n}, n \ge 1$

$\frac{dy}{dx}=-nx^{-n-1}$

$\frac{d^2y}{dx^2}=(n+1)nx^{-n-2}$

$\frac{d^3y}{dx^3}=-(n+2)(n+1)nx^{-n-3}$

let $m \in \mathbb N$

By induction, if m is even, then the derivative is positive. If m is odd, the derivative is negative

Then

$\frac{d^my}{dx^m}=\displaystyle \prod_{k=0}^{m-1}(n+k)(-1)^mx^{-n-m}$