Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Part D
![2. Write the proofs for the following derivative rules.
a.
b.
C.
d.
The derivative of a power function:
Note: the power rule works for any real n.
The derivative of the natural exponential function:
The derivative of a sum of functions:
d
dx
-(x")=;
[f(x) + g(x)] =
f(x)+g(x)]=
)=x²-1
(e) = e
On short, (x)' = nxx"
On short, (e*)' = e²|
d
f(x)+g(x)
On short, (f+g)' = f'+g'
The derivative of a difference of functions: f(x)-g(x)]=f(x)-8(x)
On short, (f-g)' = f'-g](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbb3cc9e6-2c8c-4abc-94f8-35121c73a83c%2F2cd338a4-eae4-4086-ad3f-a6314a5a0626%2Fjn6p3u6_processed.png&w=3840&q=75)
Transcribed Image Text:2. Write the proofs for the following derivative rules.
a.
b.
C.
d.
The derivative of a power function:
Note: the power rule works for any real n.
The derivative of the natural exponential function:
The derivative of a sum of functions:
d
dx
-(x")=;
[f(x) + g(x)] =
f(x)+g(x)]=
)=x²-1
(e) = e
On short, (x)' = nxx"
On short, (e*)' = e²|
d
f(x)+g(x)
On short, (f+g)' = f'+g'
The derivative of a difference of functions: f(x)-g(x)]=f(x)-8(x)
On short, (f-g)' = f'-g
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