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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![**Topic: Finding the Derivative Using Rules of Differentiation**
To find the derivative of a given function, we use standard rules of differentiation. Below is an example function:
\[ g(x) = \frac{5x^6 (x^3 - x + 3)}{x^2 + 1} \]
**Objective:**
Utilize differentiation techniques to find \( g'(x) \), the derivative of the function \( g(x) \).
**Procedure:**
1. **Apply the Quotient Rule:**
- The function \( g(x) \) is a quotient of two expressions, indicating that the quotient rule should be used. The quotient rule states:
\[
\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}
\]
where \( u = 5x^6 (x^3 - x + 3) \) and \( v = x^2 + 1 \).
2. **Differentiate the Numerator and Denominator:**
- Find the derivative of the numerator \( u \) and the denominator \( v \).
- Utilize additional differentiation rules as necessary for products or sums within \( u \).
3. **Combine Results:**
- Substitute \( u' \), \( v \), \( v' \), and \( u \) into the quotient rule formula to obtain \( g'(x) \).
This step-by-step approach helps find the derivative of complex rational functions. More specific guidance on differentiation techniques may be found in calculus textbooks and educational resources.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F310a3344-537a-48c1-a1a5-be9fa2e55789%2F9c6ffa1b-c3b1-4b73-a43e-61349d443f84%2Ffub8ub_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Finding the Derivative Using Rules of Differentiation**
To find the derivative of a given function, we use standard rules of differentiation. Below is an example function:
\[ g(x) = \frac{5x^6 (x^3 - x + 3)}{x^2 + 1} \]
**Objective:**
Utilize differentiation techniques to find \( g'(x) \), the derivative of the function \( g(x) \).
**Procedure:**
1. **Apply the Quotient Rule:**
- The function \( g(x) \) is a quotient of two expressions, indicating that the quotient rule should be used. The quotient rule states:
\[
\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}
\]
where \( u = 5x^6 (x^3 - x + 3) \) and \( v = x^2 + 1 \).
2. **Differentiate the Numerator and Denominator:**
- Find the derivative of the numerator \( u \) and the denominator \( v \).
- Utilize additional differentiation rules as necessary for products or sums within \( u \).
3. **Combine Results:**
- Substitute \( u' \), \( v \), \( v' \), and \( u \) into the quotient rule formula to obtain \( g'(x) \).
This step-by-step approach helps find the derivative of complex rational functions. More specific guidance on differentiation techniques may be found in calculus textbooks and educational resources.
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