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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![### Derivative Problem: Finding y'
In this problem, you are tasked with finding the derivative of the function \( y \) with respect to \( x \). The function is given by:
\[ y = x^3 + \sin(x) \]
#### Solution:
To find \( y' \) (which represents the derivative of \( y \) with respect to \( x \)), we need to use the rules of differentiation:
1. **Power Rule**: The derivative of \( x^n \) is \( nx^{n-1} \).
2. **Derivative of Sine Function**: The derivative of \( \sin(x) \) is \( \cos(x) \).
Applying these rules:
- The derivative of \( x^3 \) is \( 3x^2 \).
- The derivative of \( \sin(x) \) is \( \cos(x) \).
Therefore,
\[ y' = \frac{dy}{dx} = 3x^2 + \cos(x) \]
In conclusion, the derivative of the given function \( y = x^3 + \sin(x) \) with respect to \( x \) is:
\[ y' = 3x^2 + \cos(x) \]
This problem demonstrates the application of basic differentiation rules to find the rate of change of a function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22969543-4b1b-47a8-98d2-f1e5e1e74392%2F46604690-31a6-4727-bae7-f58355ac532b%2F6mg9bg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Derivative Problem: Finding y'
In this problem, you are tasked with finding the derivative of the function \( y \) with respect to \( x \). The function is given by:
\[ y = x^3 + \sin(x) \]
#### Solution:
To find \( y' \) (which represents the derivative of \( y \) with respect to \( x \)), we need to use the rules of differentiation:
1. **Power Rule**: The derivative of \( x^n \) is \( nx^{n-1} \).
2. **Derivative of Sine Function**: The derivative of \( \sin(x) \) is \( \cos(x) \).
Applying these rules:
- The derivative of \( x^3 \) is \( 3x^2 \).
- The derivative of \( \sin(x) \) is \( \cos(x) \).
Therefore,
\[ y' = \frac{dy}{dx} = 3x^2 + \cos(x) \]
In conclusion, the derivative of the given function \( y = x^3 + \sin(x) \) with respect to \( x \) is:
\[ y' = 3x^2 + \cos(x) \]
This problem demonstrates the application of basic differentiation rules to find the rate of change of a function.
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