Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of f, f', and f". f(x) = 3x - 4x5/6 f'(x) f"(x)
Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of f, f', and f". f(x) = 3x - 4x5/6 f'(x) f"(x)
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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![### Finding the First and Second Derivatives of a Function
#### Task:
Given the function \( f(x) = 3x - 4x^{5/6} \), find the first and second derivatives. Verify that your answers are reasonable by comparing the graphs of \( f(x) \), \( f'(x) \), and \( f''(x) \).
#### Function:
\[ f(x) = 3x - 4x^{5/6} \]
#### First Derivative:
\[ f'(x) = \]
*(Enter your answer in the provided box)*
#### Second Derivative:
\[ f''(x) = \]
*(Enter your answer in the provided box)*
#### Additional Resources:
If you need help, you can click on the following options:
- **Read It:** This option likely leads to a textual explanation or guide for understanding derivatives and how to compute them.
- **Watch It:** This option probably provides a video tutorial explaining the concept of differentiation and the steps to find the first and second derivatives of a function.
#### Diagram Notes:
There are no graphs or diagrams included in the current image. However, it’s suggested that for verification purposes, one would graph the function \( f(x) \) and its derivatives \( f'(x) \) and \( f''(x) \).
### Explanation:
1. **First Derivative**: This represents the rate of change of the function \( f(x) \). It is calculated by applying the power rule to each term in the function.
2. **Second Derivative**: This measures the concavity of the function \( f(x) \). It is the derivative of \( f'(x) \), indicating how the rate of change of \( f(x) \) itself is changing.
### Example Computations:
Let's proceed with computing the derivatives for the provided function:
#### First Derivative:
\[ f(x) = 3x - 4x^{5/6} \]
Applying the power rule:
\[ f'(x) = \frac{d}{dx}(3x) - \frac{d}{dx}(4x^{5/6}) \]
\[ f'(x) = 3 - \frac{5}{6} \cdot 4x^{(5/6)-1} \]
\[ f'(x) = 3 - \frac{20}{6}x^{-1/6} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff77aaeb6-32e9-4475-82f9-455a261f2944%2Fd6518970-0966-468f-b3f1-c6cf7ced795b%2F7ugnsr2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the First and Second Derivatives of a Function
#### Task:
Given the function \( f(x) = 3x - 4x^{5/6} \), find the first and second derivatives. Verify that your answers are reasonable by comparing the graphs of \( f(x) \), \( f'(x) \), and \( f''(x) \).
#### Function:
\[ f(x) = 3x - 4x^{5/6} \]
#### First Derivative:
\[ f'(x) = \]
*(Enter your answer in the provided box)*
#### Second Derivative:
\[ f''(x) = \]
*(Enter your answer in the provided box)*
#### Additional Resources:
If you need help, you can click on the following options:
- **Read It:** This option likely leads to a textual explanation or guide for understanding derivatives and how to compute them.
- **Watch It:** This option probably provides a video tutorial explaining the concept of differentiation and the steps to find the first and second derivatives of a function.
#### Diagram Notes:
There are no graphs or diagrams included in the current image. However, it’s suggested that for verification purposes, one would graph the function \( f(x) \) and its derivatives \( f'(x) \) and \( f''(x) \).
### Explanation:
1. **First Derivative**: This represents the rate of change of the function \( f(x) \). It is calculated by applying the power rule to each term in the function.
2. **Second Derivative**: This measures the concavity of the function \( f(x) \). It is the derivative of \( f'(x) \), indicating how the rate of change of \( f(x) \) itself is changing.
### Example Computations:
Let's proceed with computing the derivatives for the provided function:
#### First Derivative:
\[ f(x) = 3x - 4x^{5/6} \]
Applying the power rule:
\[ f'(x) = \frac{d}{dx}(3x) - \frac{d}{dx}(4x^{5/6}) \]
\[ f'(x) = 3 - \frac{5}{6} \cdot 4x^{(5/6)-1} \]
\[ f'(x) = 3 - \frac{20}{6}x^{-1/6} \]
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