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
![## Practice Problem: Finding the Derivative
**Problem Statement:**
Find the derivative of \( y = 7x^9 + 9^x + 3 \).
**Solution:**
\[ \frac{dy}{dx} = \]
You can use this box to input your answer. Remember to apply the power rule and the exponential rule to differentiate each term separately.
---
### Explanation:
1. For the term \(7x^9\):
- Use the power rule: \(\frac{d}{dx}[x^n] = nx^{n-1}\).
- Therefore, \(\frac{d}{dx}[7x^9] = 7 \cdot 9x^{8} = 63x^{8}\).
2. For the term \(9^x\):
- Recognize this as an exponential function. Use the rule: \(\frac{d}{dx}[a^x] = a^x \ln(a)\).
- Therefore, \(\frac{d}{dx}[9^x] = 9^x \ln(9)\).
3. For the constant term \(3\):
- The derivative of a constant is zero. So, \(\frac{d}{dx}[3] = 0\).
Combining these results:
\[ \frac{dy}{dx} = 63x^8 + 9^x \ln(9) + 0. \]
Thus, the final answer is:
\[ \frac{dy}{dx} = 63x^8 + 9^x \ln(9). \]
Feel free to enter this solution into the provided input box!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1b16e79-59da-42ec-8251-0c12184e0a83%2Fee9d0e01-748a-4a37-9117-1702f9d66228%2Fbqd3u7q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Practice Problem: Finding the Derivative
**Problem Statement:**
Find the derivative of \( y = 7x^9 + 9^x + 3 \).
**Solution:**
\[ \frac{dy}{dx} = \]
You can use this box to input your answer. Remember to apply the power rule and the exponential rule to differentiate each term separately.
---
### Explanation:
1. For the term \(7x^9\):
- Use the power rule: \(\frac{d}{dx}[x^n] = nx^{n-1}\).
- Therefore, \(\frac{d}{dx}[7x^9] = 7 \cdot 9x^{8} = 63x^{8}\).
2. For the term \(9^x\):
- Recognize this as an exponential function. Use the rule: \(\frac{d}{dx}[a^x] = a^x \ln(a)\).
- Therefore, \(\frac{d}{dx}[9^x] = 9^x \ln(9)\).
3. For the constant term \(3\):
- The derivative of a constant is zero. So, \(\frac{d}{dx}[3] = 0\).
Combining these results:
\[ \frac{dy}{dx} = 63x^8 + 9^x \ln(9) + 0. \]
Thus, the final answer is:
\[ \frac{dy}{dx} = 63x^8 + 9^x \ln(9). \]
Feel free to enter this solution into the provided input box!
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