P c=0 and I= 1. d (a) Use the definition of the derivative to prove IP d. -I" = nr"-1 for all n E No. dr (b) Use induction and the Product Rule to prove (c) Use the Quotient Rule to prove d. = nr"-1 for all z ER\{0}, n EZ\No. 1 T1/n = dr (d) Use the Inverse Function Theorem to prove -/n-1 for all zE (0, 00), n E N. d m (e) Use the Chain Rule to prove dr Tm/n-1 for all r E (0, 00), m e Z, ne N. (f) Use the fact that if {tn} is a sequence in R, then lim n+00 daT d lim r to prove that dr txt-1 dr n+0 for all r E (0, 00), t €R\Q.

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