D Transcribed Image Text: (a) Use the definition of the derivative to prove C=0 and da T = 1. %3D (b)

Excerpt
(0, 00), m E Z, n E N. %D – dr tn (f) Use the fact that if {tn} is a sequence in R, then lim lim r to prove that dr = trt-1 for all r E (0, o), t E R \ Q.

D Transcribed Image Text: (a) Use the definition of the derivative to prove
C=0 and
da
T = 1.
%3D
(b) Use induction and the Product Rule to prove
I = nI”-1 for all n E No-
(c) Use the Quotient Rule to prove
= na”-1 for all I ER\{0}, n E Z\ No.
dr
d
(d) Use the Inverse Function Theorem to prove
d.x
1
= =r/n-1 for all r E (0, 00), n E N.
d.
(e) Use the Chain Rule to prove
m/n
/m-1 for all x E (0, 00), m E Z, n E N.
%D

dr
tn
(f) Use the fact that if {tn} is a sequence in R, then lim
lim r to prove
that
dr
= trt-1
for all r E (0, o), t E R \ Q.

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