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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![### Problem Statement
Differentiate the following function:
\[ h(w) = (w^5 + 8w^4)(w^{-4} - w^{-6}) \]
### Explanation
This expression is the product of two polynomial functions. To differentiate \( h(w) \), you can apply the product rule. The product rule is expressed as:
\[ (uv)' = u'v + uv' \]
where \( u \) and \( v \) are functions of \( w \).
### Steps for Solution
1. **Identify \( u \) and \( v \):**
- \( u = w^5 + 8w^4 \)
- \( v = w^{-4} - w^{-6} \)
2. **Find the derivatives \( u' \) and \( v' \):**
- \( u' = \frac{d}{dw}(w^5 + 8w^4) = 5w^4 + 32w^3 \)
- \( v' = \frac{d}{dw}(w^{-4} - w^{-6}) = -4w^{-5} + 6w^{-7} \)
3. **Apply the product rule:**
- \( h'(w) = u'v + uv' \)
- Substitute back into the derivative formula and simplify if necessary.
This approach provides the step-by-step process to differentiate the given function using the appropriate rules of calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4566d3ec-93a4-4a71-af5b-e7d0422050f1%2F8ab3de67-a4e8-4ce7-b8c4-42675b39e52c%2Fv73x8h9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Differentiate the following function:
\[ h(w) = (w^5 + 8w^4)(w^{-4} - w^{-6}) \]
### Explanation
This expression is the product of two polynomial functions. To differentiate \( h(w) \), you can apply the product rule. The product rule is expressed as:
\[ (uv)' = u'v + uv' \]
where \( u \) and \( v \) are functions of \( w \).
### Steps for Solution
1. **Identify \( u \) and \( v \):**
- \( u = w^5 + 8w^4 \)
- \( v = w^{-4} - w^{-6} \)
2. **Find the derivatives \( u' \) and \( v' \):**
- \( u' = \frac{d}{dw}(w^5 + 8w^4) = 5w^4 + 32w^3 \)
- \( v' = \frac{d}{dw}(w^{-4} - w^{-6}) = -4w^{-5} + 6w^{-7} \)
3. **Apply the product rule:**
- \( h'(w) = u'v + uv' \)
- Substitute back into the derivative formula and simplify if necessary.
This approach provides the step-by-step process to differentiate the given function using the appropriate rules of calculus.
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