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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![### Using the Quotient Rule to Calculate the Derivative
To calculate the derivative of the function \( f(x) = \frac{8}{1+e^x} \) using the Quotient Rule, follow these steps:
The Quotient Rule is given by:
\[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]
Where \( u \) and \( v \) are functions of \( x \), and \( u' \) and \( v' \) are their respective derivatives.
For the given function:
- \( u = 8 \)
- \( v = 1 + e^x \)
The derivatives of \( u \) and \( v \) are:
- \( u' = 0 \) (since the derivative of a constant is zero)
- \( v' = e^x \) (since the derivative of \( e^x \) is \( e^x \))
Substituting these into the Quotient Rule formula, we get:
\[ f'(x) = \frac{(0)(1 + e^x) - (8)(e^x)}{(1 + e^x)^2} \]
\[ f'(x) = \frac{0 - 8e^x}{(1 + e^x)^2} \]
\[ f'(x) = \frac{-8e^x}{(1 + e^x)^2} \]
Thus, the derivative \( f'(x) \) is:
\[ f'(x) = \boxed{\frac{-8e^x}{(1 + e^x)^2}} \]
This employs symbolic notation and fractions to maintain mathematical rigor and clarity in the derivation process.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2d79032-32db-4435-bc8c-34ca2691b1b6%2F6db21fe4-dc95-4af2-979c-2d736302bcda%2Fc58t7ge_processed.png&w=3840&q=75)
Transcribed Image Text:### Using the Quotient Rule to Calculate the Derivative
To calculate the derivative of the function \( f(x) = \frac{8}{1+e^x} \) using the Quotient Rule, follow these steps:
The Quotient Rule is given by:
\[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]
Where \( u \) and \( v \) are functions of \( x \), and \( u' \) and \( v' \) are their respective derivatives.
For the given function:
- \( u = 8 \)
- \( v = 1 + e^x \)
The derivatives of \( u \) and \( v \) are:
- \( u' = 0 \) (since the derivative of a constant is zero)
- \( v' = e^x \) (since the derivative of \( e^x \) is \( e^x \))
Substituting these into the Quotient Rule formula, we get:
\[ f'(x) = \frac{(0)(1 + e^x) - (8)(e^x)}{(1 + e^x)^2} \]
\[ f'(x) = \frac{0 - 8e^x}{(1 + e^x)^2} \]
\[ f'(x) = \frac{-8e^x}{(1 + e^x)^2} \]
Thus, the derivative \( f'(x) \) is:
\[ f'(x) = \boxed{\frac{-8e^x}{(1 + e^x)^2}} \]
This employs symbolic notation and fractions to maintain mathematical rigor and clarity in the derivation process.
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