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...
Related questions
Question
![This image appears to show a handwritten mathematical derivation involving calculus. Here is a detailed transcription suitable for an educational website:
---
### Calculating the Second Derivative of a Function
Consider the function \[ y = x \] raised to the power of \( x \). We want to find the second derivative, \[ \frac{d^2}{dx^2}(\frac{1 - x}{x}). \]
Starting with the first step:
\[ e^{ } \hspace{1em} \]
Now, consider the derivative of the expression inside:
\[ \left( \frac{1 - x}{x} \right) \]
First, simplify the expression:
\[ \frac{1}{x} - 1 \]
Differentiate it with respect to \( x \):
\[ - \frac{1}{x^2} \]
Then, take the second derivative:
\[ \frac{d^2}{dx^2} \left( \frac{1 - x}{x} \right) = -\frac{d^2}{dx^2} \left( \frac{1}{x} \right) \]
where:
\[ \frac{d^2}{dx^2} \left( \frac{1}{x} \right) = \frac{2}{x^3} \]
Putting it all together, we obtain:
\[ -\frac{2}{x^3} \]
Therefore, the second derivative of the function is \(-\frac{2}{x^3}\).
---
The handwritten section seems to include some steps and elements to explain the differentiation process. This derivation demonstrates the application of basic calculus techniques to find higher-order derivatives. No graphs or diagrams are included in the image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2c89fb8-28c7-4f54-82f8-eda7977d51b8%2F73b92119-0595-4399-8072-2a8dad005553%2Fdsldpty_processed.jpeg&w=3840&q=75)
Transcribed Image Text:This image appears to show a handwritten mathematical derivation involving calculus. Here is a detailed transcription suitable for an educational website:
---
### Calculating the Second Derivative of a Function
Consider the function \[ y = x \] raised to the power of \( x \). We want to find the second derivative, \[ \frac{d^2}{dx^2}(\frac{1 - x}{x}). \]
Starting with the first step:
\[ e^{ } \hspace{1em} \]
Now, consider the derivative of the expression inside:
\[ \left( \frac{1 - x}{x} \right) \]
First, simplify the expression:
\[ \frac{1}{x} - 1 \]
Differentiate it with respect to \( x \):
\[ - \frac{1}{x^2} \]
Then, take the second derivative:
\[ \frac{d^2}{dx^2} \left( \frac{1 - x}{x} \right) = -\frac{d^2}{dx^2} \left( \frac{1}{x} \right) \]
where:
\[ \frac{d^2}{dx^2} \left( \frac{1}{x} \right) = \frac{2}{x^3} \]
Putting it all together, we obtain:
\[ -\frac{2}{x^3} \]
Therefore, the second derivative of the function is \(-\frac{2}{x^3}\).
---
The handwritten section seems to include some steps and elements to explain the differentiation process. This derivation demonstrates the application of basic calculus techniques to find higher-order derivatives. No graphs or diagrams are included in the image.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning