Suppose n is a negative integer and f(x) = xⁿ. Use the following steps to prove that f'(a) = na^{n - 1}, which means the Power Rule for positive integers extends to all integers.

a. Assume that m = - n, so that m > 0. Use the definition

Simplify using the factoring rule (which is valid for n > 0)

b. Use this result to find d/dx (x^-7) and d/dx (1/x¹⁰). 

Transcribed Image Text:

x" = lim х—а х — а a" т х lim х — а f' (a) ха х" — а" — (х — а)(x" -1 + x" ?а + ... a)(x"-1 + x"¯²a + · .. + xa"-2 + a"-1) in