(f) Use the fact that if {tn} is a sequence in R, then lim n+0o d.r d lim r to prove that da n-0 d. r' = txt-1 dr !! for all a E (0, o0), tER\ Q.
(f) Use the fact that if {tn} is a sequence in R, then lim n+0o d.r d lim r to prove that da n-0 d. r' = txt-1 dr !! for all a E (0, o0), tER\ Q.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
F please
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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