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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![### Calculus Problem: Finding the Derivative
**Problem Statement:**
7. Find \( y' \) if \( y = x^3 + \sin(x) \).
**Solution Guide:**
1. **Identify the Function:**
The given function is \( y = x^3 + \sin(x) \).
2. **Apply the Derivative Rules:**
We will use the power rule for \( x^3 \) and the derivative rule for \( \sin(x) \).
- The power rule states that if \( y = x^n \), then \( \frac{dy}{dx} = nx^{n-1} \).
- The derivative of \( \sin(x) \) is \( \cos(x) \).
3. **Differentiate Each Term:**
- For \( x^3 \):
\[
\frac{d}{dx}(x^3) = 3x^2
\]
- For \( \sin(x) \):
\[
\frac{d}{dx}(\sin(x)) = \cos(x)
\]
4. **Combine the Results:**
The derivative \( y' \) is the sum of the derivatives of the individual terms.
Therefore:
\[
y' = 3x^2 + \cos(x)
\]
5. **Final Answer:**
\[
\boxed{y' = 3x^2 + \cos(x)}
\]
By following these steps, we have found the derivative \( y' \) of the given function \( y = x^3 + \sin(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22969543-4b1b-47a8-98d2-f1e5e1e74392%2F833390c6-05b4-46dc-a7cd-37a06b0bf38d%2Fzjuir7b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus Problem: Finding the Derivative
**Problem Statement:**
7. Find \( y' \) if \( y = x^3 + \sin(x) \).
**Solution Guide:**
1. **Identify the Function:**
The given function is \( y = x^3 + \sin(x) \).
2. **Apply the Derivative Rules:**
We will use the power rule for \( x^3 \) and the derivative rule for \( \sin(x) \).
- The power rule states that if \( y = x^n \), then \( \frac{dy}{dx} = nx^{n-1} \).
- The derivative of \( \sin(x) \) is \( \cos(x) \).
3. **Differentiate Each Term:**
- For \( x^3 \):
\[
\frac{d}{dx}(x^3) = 3x^2
\]
- For \( \sin(x) \):
\[
\frac{d}{dx}(\sin(x)) = \cos(x)
\]
4. **Combine the Results:**
The derivative \( y' \) is the sum of the derivatives of the individual terms.
Therefore:
\[
y' = 3x^2 + \cos(x)
\]
5. **Final Answer:**
\[
\boxed{y' = 3x^2 + \cos(x)}
\]
By following these steps, we have found the derivative \( y' \) of the given function \( y = x^3 + \sin(x) \).
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