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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Please include step by step on paper and circle answer. Thank you.
![**Find the derivative of the function.**
Given the function:
\[ s = t^7 e^t \]
You are required to find the derivative of this function with respect to \( t \). Remember to use the product rule for differentiation since the function is a product of \( t^7 \) and \( e^t \).
**Differentiation Instruction:**
1. **Identify the Functions:**
- Let \( u(t) = t^7 \)
- Let \( v(t) = e^t \)
2. **Apply the Product Rule:**
- The product rule states that if \( s = u \cdot v \), then \( s' = u' \cdot v + u \cdot v' \).
3. **Differentiate Each Function:**
- \( u'(t) = \frac{d}{dt}(t^7) = 7t^6 \)
- \( v'(t) = \frac{d}{dt}(e^t) = e^t \)
4. **Substitute into the Product Rule:**
- \( s' = (7t^6) \cdot (e^t) + (t^7) \cdot (e^t) \)
5. **Simplify the Expression:**
- \( s' = 7t^6 e^t + t^7 e^t \)
6. **Factor the Expression:**
- \( s' = t^6 e^t (7 + t) \)
This is the derivative of the given function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F125111cc-25a5-4f05-886d-ef1bfa459fc8%2F89f0d86b-b8e6-4804-ad82-ee6b3791798d%2Ftk0vr2p_processed.png&w=3840&q=75)
Transcribed Image Text:**Find the derivative of the function.**
Given the function:
\[ s = t^7 e^t \]
You are required to find the derivative of this function with respect to \( t \). Remember to use the product rule for differentiation since the function is a product of \( t^7 \) and \( e^t \).
**Differentiation Instruction:**
1. **Identify the Functions:**
- Let \( u(t) = t^7 \)
- Let \( v(t) = e^t \)
2. **Apply the Product Rule:**
- The product rule states that if \( s = u \cdot v \), then \( s' = u' \cdot v + u \cdot v' \).
3. **Differentiate Each Function:**
- \( u'(t) = \frac{d}{dt}(t^7) = 7t^6 \)
- \( v'(t) = \frac{d}{dt}(e^t) = e^t \)
4. **Substitute into the Product Rule:**
- \( s' = (7t^6) \cdot (e^t) + (t^7) \cdot (e^t) \)
5. **Simplify the Expression:**
- \( s' = 7t^6 e^t + t^7 e^t \)
6. **Factor the Expression:**
- \( s' = t^6 e^t (7 + t) \)
This is the derivative of the given function.
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