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
![**Title: Differentiating Functions Using the Product Rule**
In this lesson, we will learn how to differentiate a given function using the product rule. The expression provided is:
\[
(2x - 4)^{1/2} (x^3 - x^{24} + 5x)
\]
**Steps to Solve:**
1. **Identify the Functions:**
The function consists of two parts:
- \( u(x) = (2x - 4)^{1/2} \)
- \( v(x) = x^3 - x^{24} + 5x \)
2. **Apply the Product Rule:**
The derivative of a product of two functions, \( u(x) \) and \( v(x) \), is given by:
\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]
3. **Differentiate Each Function:**
- Find \( u'(x) \): Differentiate \( (2x - 4)^{1/2} \).
- Find \( v'(x) \): Differentiate \( x^3 - x^{24} + 5x \).
4. **Substitute and Simplify:**
- Substitute \( u'(x) \) and \( v'(x) \) into the product rule formula.
- Simplify the resulting expression to obtain the derivative.
By following these steps, students will be able to differentiate complex functions using the product rule effectively. Remember, careful differentiation and substitution are key to obtaining a correct result.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2d8bd87-a7b3-4179-b62c-44ece0415739%2F413b4e82-ba07-4b80-b2e2-4d759177e23c%2Fg9metft_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Differentiating Functions Using the Product Rule**
In this lesson, we will learn how to differentiate a given function using the product rule. The expression provided is:
\[
(2x - 4)^{1/2} (x^3 - x^{24} + 5x)
\]
**Steps to Solve:**
1. **Identify the Functions:**
The function consists of two parts:
- \( u(x) = (2x - 4)^{1/2} \)
- \( v(x) = x^3 - x^{24} + 5x \)
2. **Apply the Product Rule:**
The derivative of a product of two functions, \( u(x) \) and \( v(x) \), is given by:
\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]
3. **Differentiate Each Function:**
- Find \( u'(x) \): Differentiate \( (2x - 4)^{1/2} \).
- Find \( v'(x) \): Differentiate \( x^3 - x^{24} + 5x \).
4. **Substitute and Simplify:**
- Substitute \( u'(x) \) and \( v'(x) \) into the product rule formula.
- Simplify the resulting expression to obtain the derivative.
By following these steps, students will be able to differentiate complex functions using the product rule effectively. Remember, careful differentiation and substitution are key to obtaining a correct result.
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