Find the derivative of F(y). Select the correct answer. F( y) = sinh y tanh y F' (y) = sinh y sech y + tanh y cosh y F' ( y) = sinh y sech 2 y + tanh y cosh y F' (y) = sinh y sech 2 y + tanh y cosh ² y 2 F'(y) = sinh y sech ² y + tanh y 2
Find the derivative of F(y). Select the correct answer. F( y) = sinh y tanh y F' (y) = sinh y sech y + tanh y cosh y F' ( y) = sinh y sech 2 y + tanh y cosh y F' (y) = sinh y sech 2 y + tanh y cosh ² y 2 F'(y) = sinh y sech ² y + tanh y 2
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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![### Derivatives of Hyperbolic Functions
**Problem:**
Find the derivative of \( F(y) \).
\[ F(y) = \sinh y \tanh y \]
**Solution:**
Select the correct answer.
1. \( F'(y) = \sinh y \, \text{sech} \, y + \tanh y \, \cosh y \)
2. \( F'(y) = \sinh y \, \text{sech}^2 \, y + \tanh y \, \cosh y \)
3. \( F'(y) = \sinh y \, \text{sech}^2 \, y + \tanh y \, \cosh^2 \, y \)
4. \( F'(y) = \sinh y \, \text{sech}^2 \, y + \tanh y \)
### Explanation:
To solve this, we need to apply the product rule for differentiation and the chain rule, as well as leverage the hyperbolic function identities. Specifically, recall the derivatives:
- \( \frac{d}{dy} (\sinh y) = \cosh y \)
- \( \frac{d}{dy} (\tanh y) = \text{sech}^2 y \)
Using these, we can find \( F'(y) \) step by step.
### Detailed Solution
1. Start with the function \( F(y) = \sinh y \tanh y \).
2. Apply the product rule:
\[ F'(y) = (\sinh y)' \tanh y + \sinh y (\tanh y)' \]
3. Note the derivatives of the component functions:
\[ (\sinh y)' = \cosh y \]
\[ (\tanh y)' = \text{sech}^2 y \]
4. Substitute these into the product rule:
\[ F'(y) = \cosh y \tanh y + \sinh y \text{sech}^2 y \]
Based on this calculation, the correct answer is:
\[ \boxed{ F'(y) = \sinh y \, \text{sech}^2 \, y + \tanh y \, }\]
Correct answer: \[ \sinh y \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24e8bfde-46ac-4815-81bc-d8be1d9c4f35%2F7a86788e-19c3-4d24-b675-f0bb54d79282%2F4utzxqf_processed.png&w=3840&q=75)
Transcribed Image Text:### Derivatives of Hyperbolic Functions
**Problem:**
Find the derivative of \( F(y) \).
\[ F(y) = \sinh y \tanh y \]
**Solution:**
Select the correct answer.
1. \( F'(y) = \sinh y \, \text{sech} \, y + \tanh y \, \cosh y \)
2. \( F'(y) = \sinh y \, \text{sech}^2 \, y + \tanh y \, \cosh y \)
3. \( F'(y) = \sinh y \, \text{sech}^2 \, y + \tanh y \, \cosh^2 \, y \)
4. \( F'(y) = \sinh y \, \text{sech}^2 \, y + \tanh y \)
### Explanation:
To solve this, we need to apply the product rule for differentiation and the chain rule, as well as leverage the hyperbolic function identities. Specifically, recall the derivatives:
- \( \frac{d}{dy} (\sinh y) = \cosh y \)
- \( \frac{d}{dy} (\tanh y) = \text{sech}^2 y \)
Using these, we can find \( F'(y) \) step by step.
### Detailed Solution
1. Start with the function \( F(y) = \sinh y \tanh y \).
2. Apply the product rule:
\[ F'(y) = (\sinh y)' \tanh y + \sinh y (\tanh y)' \]
3. Note the derivatives of the component functions:
\[ (\sinh y)' = \cosh y \]
\[ (\tanh y)' = \text{sech}^2 y \]
4. Substitute these into the product rule:
\[ F'(y) = \cosh y \tanh y + \sinh y \text{sech}^2 y \]
Based on this calculation, the correct answer is:
\[ \boxed{ F'(y) = \sinh y \, \text{sech}^2 \, y + \tanh y \, }\]
Correct answer: \[ \sinh y \
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