Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![**Problem Statement:**
Given
\[ f(t) = \frac{t}{t + 1} \]
determine \( f''(t) \) when \( t = 1 \).
**Note:** Enter your answer as a decimal.
[Text Box for Answer]
---
**Explanation:**
To solve this problem, follow these steps:
1. **First Derivative:**
Calculate the first derivative \( f'(t) \) of the function using the quotient rule:
\[ f(t) = \frac{t}{t + 1} \]
\[ f'(t) = \frac{d}{dt} \left( \frac{t}{t + 1} \right) \]
Use the quotient rule for derivatives \(\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \), where \( u = t \) and \( v = t + 1 \).
2. **Second Derivative:**
Determine the second derivative \( f''(t) \) using differentiation rules on the first derivative found in the previous step.
3. **Evaluate at \( t = 1 \):**
Substitute \( t = 1 \) into the second derivative \( f''(t) \) to get the specific value.
4. **Express as a Decimal:**
Convert the result of \( f''(t) \) at \( t = 1 \) to a decimal form.
This process requires knowledge of calculus, particularly differentiation techniques like the quotient rule. Ensure to simplify each step to avoid common mistakes.
After solving, input the decimal result in the provided text box for the answer.
---
Try to derive the answer step-by-step to reinforce your understanding of differentiation.](https://content.bartleby.com/qna-images/question/8f96e7e7-1e0c-47dd-ab8a-07b078e3d874/959a8b0d-2b3d-4a23-a612-8139c54017e2/2wo6db.png)
Transcribed Image Text:**Problem Statement:**
Given
\[ f(t) = \frac{t}{t + 1} \]
determine \( f''(t) \) when \( t = 1 \).
**Note:** Enter your answer as a decimal.
[Text Box for Answer]
---
**Explanation:**
To solve this problem, follow these steps:
1. **First Derivative:**
Calculate the first derivative \( f'(t) \) of the function using the quotient rule:
\[ f(t) = \frac{t}{t + 1} \]
\[ f'(t) = \frac{d}{dt} \left( \frac{t}{t + 1} \right) \]
Use the quotient rule for derivatives \(\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \), where \( u = t \) and \( v = t + 1 \).
2. **Second Derivative:**
Determine the second derivative \( f''(t) \) using differentiation rules on the first derivative found in the previous step.
3. **Evaluate at \( t = 1 \):**
Substitute \( t = 1 \) into the second derivative \( f''(t) \) to get the specific value.
4. **Express as a Decimal:**
Convert the result of \( f''(t) \) at \( t = 1 \) to a decimal form.
This process requires knowledge of calculus, particularly differentiation techniques like the quotient rule. Ensure to simplify each step to avoid common mistakes.
After solving, input the decimal result in the provided text box for the answer.
---
Try to derive the answer step-by-step to reinforce your understanding of differentiation.
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