College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter4: Exponential And Logarithmic Functions
Section: Chapter Questions
Problem 3CC: If xis large, which function grows faster, f(x)=2x or g(x)=x2?
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![**Problem Statement:**
If \( f(x) = \frac{6 - x^2}{5 + x^2} \), find:
\[ f'(x) = \]
**Explanation:**
To find the derivative \( f'(x) \) of the given function, you can apply the Quotient Rule. The function \( f(x) \) is presented as a fraction with numerator \( 6 - x^2 \) and denominator \( 5 + x^2 \).
**Quotient Rule**: If you have a function \( f(x) = \frac{u(x)}{v(x)} \), its derivative \( f'(x) \) is given by:
\[
f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}
\]
Here, \( u(x) = 6 - x^2 \) and \( v(x) = 5 + x^2 \).
To find \( f'(x) \), calculate:
1. \( u'(x) = \) derivative of the numerator \( 6 - x^2 \).
2. \( v'(x) = \) derivative of the denominator \( 5 + x^2 \).
Substitute these derivatives into the Quotient Rule formula to find \( f'(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca24997c-457e-453a-a19b-7a30b4d1b1d9%2Fbe221a72-867e-4aa7-9863-de3f4ca29cf6%2Fi17fhfp_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
If \( f(x) = \frac{6 - x^2}{5 + x^2} \), find:
\[ f'(x) = \]
**Explanation:**
To find the derivative \( f'(x) \) of the given function, you can apply the Quotient Rule. The function \( f(x) \) is presented as a fraction with numerator \( 6 - x^2 \) and denominator \( 5 + x^2 \).
**Quotient Rule**: If you have a function \( f(x) = \frac{u(x)}{v(x)} \), its derivative \( f'(x) \) is given by:
\[
f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}
\]
Here, \( u(x) = 6 - x^2 \) and \( v(x) = 5 + x^2 \).
To find \( f'(x) \), calculate:
1. \( u'(x) = \) derivative of the numerator \( 6 - x^2 \).
2. \( v'(x) = \) derivative of the denominator \( 5 + x^2 \).
Substitute these derivatives into the Quotient Rule formula to find \( f'(x) \).
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