### Calculating Derivatives in Advanced Mathematics **Problem Statement:** We are tasked with calculating the derivative \( f''(1) \) for the given function \( f(t) = \frac{5t}{t+1} \). **Function Definition:** \[ f(t) = \frac{5t}{t+1} \] **Instructions:** - Provide an exact answer. - Use symbolic notation and fractions where necessary. **Student Submission:** \[ f''(1) = \frac{5}{4} \] Status: **Incorrect** --- **Detailed Explanation:** Let's dive deeper into solving this problem and understand why the given answer might be incorrect. 1. **First Derivative Calculation:** To find \( f''(1) \), we first need to determine the first derivative of \( f(t) \). Using the quotient rule for differentiation \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \), where \( u = 5t \) and \( v = t + 1 \): \[ u' = 5, \quad v' = 1 \] \[ f'(t) = \frac{(5)(t+1) - (5t)(1)}{(t + 1)^2} \] Simplify the numerator: \[ f'(t) = \frac{5t + 5 - 5t}{(t + 1)^2} = \frac{5}{(t + 1)^2} \] 2. **Second Derivative Calculation:** Next, we find the second derivative \( f''(t) \): \[ f''(t) = \left( \frac{5}{(t + 1)^2} \right)' \] Applying the chain rule: \[ f''(t) = 5 \cdot \left( (t + 1)^{-2} \right)' \] The derivative of \( (t + 1)^{-2} \): \[ (t + 1)^{-2} = -2(t + 1)^{-3} \cdot (1) = -\frac{2}{(t + 1)^3}
### Calculating Derivatives in Advanced Mathematics **Problem Statement:** We are tasked with calculating the derivative \( f''(1) \) for the given function \( f(t) = \frac{5t}{t+1} \). **Function Definition:** \[ f(t) = \frac{5t}{t+1} \] **Instructions:** - Provide an exact answer. - Use symbolic notation and fractions where necessary. **Student Submission:** \[ f''(1) = \frac{5}{4} \] Status: **Incorrect** --- **Detailed Explanation:** Let's dive deeper into solving this problem and understand why the given answer might be incorrect. 1. **First Derivative Calculation:** To find \( f''(1) \), we first need to determine the first derivative of \( f(t) \). Using the quotient rule for differentiation \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \), where \( u = 5t \) and \( v = t + 1 \): \[ u' = 5, \quad v' = 1 \] \[ f'(t) = \frac{(5)(t+1) - (5t)(1)}{(t + 1)^2} \] Simplify the numerator: \[ f'(t) = \frac{5t + 5 - 5t}{(t + 1)^2} = \frac{5}{(t + 1)^2} \] 2. **Second Derivative Calculation:** Next, we find the second derivative \( f''(t) \): \[ f''(t) = \left( \frac{5}{(t + 1)^2} \right)' \] Applying the chain rule: \[ f''(t) = 5 \cdot \left( (t + 1)^{-2} \right)' \] The derivative of \( (t + 1)^{-2} \): \[ (t + 1)^{-2} = -2(t + 1)^{-3} \cdot (1) = -\frac{2}{(t + 1)^3}
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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![### Calculating Derivatives in Advanced Mathematics
**Problem Statement:**
We are tasked with calculating the derivative \( f''(1) \) for the given function \( f(t) = \frac{5t}{t+1} \).
**Function Definition:**
\[ f(t) = \frac{5t}{t+1} \]
**Instructions:**
- Provide an exact answer.
- Use symbolic notation and fractions where necessary.
**Student Submission:**
\[ f''(1) = \frac{5}{4} \]
Status: **Incorrect**
---
**Detailed Explanation:**
Let's dive deeper into solving this problem and understand why the given answer might be incorrect.
1. **First Derivative Calculation:**
To find \( f''(1) \), we first need to determine the first derivative of \( f(t) \).
Using the quotient rule for differentiation \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \), where \( u = 5t \) and \( v = t + 1 \):
\[
u' = 5, \quad v' = 1
\]
\[
f'(t) = \frac{(5)(t+1) - (5t)(1)}{(t + 1)^2}
\]
Simplify the numerator:
\[
f'(t) = \frac{5t + 5 - 5t}{(t + 1)^2} = \frac{5}{(t + 1)^2}
\]
2. **Second Derivative Calculation:**
Next, we find the second derivative \( f''(t) \):
\[
f''(t) = \left( \frac{5}{(t + 1)^2} \right)'
\]
Applying the chain rule:
\[
f''(t) = 5 \cdot \left( (t + 1)^{-2} \right)'
\]
The derivative of \( (t + 1)^{-2} \):
\[
(t + 1)^{-2} = -2(t + 1)^{-3} \cdot (1) = -\frac{2}{(t + 1)^3}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F575d721b-26eb-4db6-af93-50c207cd3fab%2F8c7bcaca-97f2-434f-a4d3-349e197359f5%2Ffd2lzd.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating Derivatives in Advanced Mathematics
**Problem Statement:**
We are tasked with calculating the derivative \( f''(1) \) for the given function \( f(t) = \frac{5t}{t+1} \).
**Function Definition:**
\[ f(t) = \frac{5t}{t+1} \]
**Instructions:**
- Provide an exact answer.
- Use symbolic notation and fractions where necessary.
**Student Submission:**
\[ f''(1) = \frac{5}{4} \]
Status: **Incorrect**
---
**Detailed Explanation:**
Let's dive deeper into solving this problem and understand why the given answer might be incorrect.
1. **First Derivative Calculation:**
To find \( f''(1) \), we first need to determine the first derivative of \( f(t) \).
Using the quotient rule for differentiation \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \), where \( u = 5t \) and \( v = t + 1 \):
\[
u' = 5, \quad v' = 1
\]
\[
f'(t) = \frac{(5)(t+1) - (5t)(1)}{(t + 1)^2}
\]
Simplify the numerator:
\[
f'(t) = \frac{5t + 5 - 5t}{(t + 1)^2} = \frac{5}{(t + 1)^2}
\]
2. **Second Derivative Calculation:**
Next, we find the second derivative \( f''(t) \):
\[
f''(t) = \left( \frac{5}{(t + 1)^2} \right)'
\]
Applying the chain rule:
\[
f''(t) = 5 \cdot \left( (t + 1)^{-2} \right)'
\]
The derivative of \( (t + 1)^{-2} \):
\[
(t + 1)^{-2} = -2(t + 1)^{-3} \cdot (1) = -\frac{2}{(t + 1)^3}
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