Prove the case r= -n (n is a positive integer)of the general power rule

[tachyon_man]

Prove that:
d/dx x^-n = -nx^-n-1
Use the factorization of a difference of nth powers given in this section (not using quotient rule)
My attempt gets me from the definition of the derivative to (1/n^(n-1)) n times... I need the negative. I get nx^(-n-1) instead of -nx^(n-1).

 

[HallsofIvy]

Prove that:
d/dx x^-n = -nx^-n-1
Use the factorization of a difference of nth powers given in this section (not using quotient rule)
My attempt gets me from the definition of the derivative to (1/n^(n-1)) n times... I need the negative. I get nx^(-n-1) instead of -nx^(n-1).

Since you chose not to say what "the factorization of a difference of nth powers given in this section" nor now you attempted to do this, I don't see how you can expect anyone to help.

Since you are not allowed to use the quotient rule, I would recommend writing $x^n(x^{-n})= 1$ and differentiating both sides, using the product rule on the left. What do you get when you do that?