(a) Prove the power rule when c is a positive rational. Suggestion: Assume c = p/q for some positive integers p and q. Define the function f(x) = xP/q. Then this function satisfies f (x)ª = x". Use implicit differentiation. (b) Prove the power rule when c is a negative rational. Suggestion: Look at what you have done so far in this assignment.
(a) Prove the power rule when c is a positive rational. Suggestion: Assume c = p/q for some positive integers p and q. Define the function f(x) = xP/q. Then this function satisfies f (x)ª = x". Use implicit differentiation. (b) Prove the power rule when c is a negative rational. Suggestion: Look at what you have done so far in this assignment.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Kindly explain
![3. The power rule says that,
for
every c E R:
d
[æ°]
dx
= crc-1
In this problem, we will restrict ourselves only to the domain x > 0.
In Video 3.7 you learned a proof for a particular case: when c is a positive integer. You will
later (Video 4.10) learn a proof that works for all c E R using logarithms, but there are other
simple proofs, without using logarithms, that extend to c E Q. That is the goal of this problem.
You may assume the power rule when c is a positive integer, as well as other results
in Unit 3, including the Chain Rule.
you
learned
(a) Prove the power rule when c is a positive rational.
Suggestion: Assume c =
f (x) = xP/4. Then this function satisfies
p/q for some positive integers p and q. Define the function
f (x)ª = x°.
Use implicit differentiation.
(b) Prove the power rule when c is a negative rational.
Suggestion: Look at what you have done so far in this assignment.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F82b5c32c-2578-4910-aa75-c8367044e495%2Fd33e48a1-10d6-443c-be72-a8d56e541e70%2Fcwmhtf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. The power rule says that,
for
every c E R:
d
[æ°]
dx
= crc-1
In this problem, we will restrict ourselves only to the domain x > 0.
In Video 3.7 you learned a proof for a particular case: when c is a positive integer. You will
later (Video 4.10) learn a proof that works for all c E R using logarithms, but there are other
simple proofs, without using logarithms, that extend to c E Q. That is the goal of this problem.
You may assume the power rule when c is a positive integer, as well as other results
in Unit 3, including the Chain Rule.
you
learned
(a) Prove the power rule when c is a positive rational.
Suggestion: Assume c =
f (x) = xP/4. Then this function satisfies
p/q for some positive integers p and q. Define the function
f (x)ª = x°.
Use implicit differentiation.
(b) Prove the power rule when c is a negative rational.
Suggestion: Look at what you have done so far in this assignment.
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