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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Question
![**Problem 8:** If \( f(x) = x^{\pi} \cosh x \), find \( f''(x) \).
**Solution:**
To find the second derivative \( f''(x) \), we start by finding the first derivative \( f'(x) \).
1. **First Derivative (\( f'(x) \))**:
- \( f(x) = x^{\pi} \cosh x \)
- Here, we need to apply the product rule: \( (uv)' = u'v + uv' \), where \( u = x^{\pi} \) and \( v = \cosh x \).
- Derivatives:
- \( u' = \pi x^{\pi-1} \)
- \( v' = \sinh x \)
- Applying the product rule:
\[
f'(x) = (\pi x^{\pi-1}) \cosh x + (x^{\pi}) \sinh x
\]
2. **Second Derivative (\( f''(x) \))**:
- Differentiate \( f'(x) \) again using the product rule on each term:
- For \( (\pi x^{\pi-1} \cosh x) \):
- Apply product rule:
\[
\left(\pi x^{\pi-1}\right)' = \pi(\pi-1)x^{\pi-2}
\]
\[
( \pi(\pi-1)x^{\pi-2} ) \cosh x + (\pi x^{\pi-1}) \sinh x
\]
- For \( (x^{\pi} \sinh x) \):
- Apply product rule:
\[
(x^{\pi})' = \pi x^{\pi-1}
\]
\[
(\pi x^{\pi-1}) \sinh x + (x^{\pi}) \cosh x
\]
- Combine terms:
\[
f''(x) = [\pi(\pi-1)x^{\pi-2} \cosh x + \pi x^{\pi-1} \sinh x] + [\pi x^{\pi-1} \sinh x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69d3c8b6-4012-43b0-80a6-d85ebf5fdc33%2Ff61cd3da-68b1-468f-85fc-71e28742ec3a%2Frhh614d_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 8:** If \( f(x) = x^{\pi} \cosh x \), find \( f''(x) \).
**Solution:**
To find the second derivative \( f''(x) \), we start by finding the first derivative \( f'(x) \).
1. **First Derivative (\( f'(x) \))**:
- \( f(x) = x^{\pi} \cosh x \)
- Here, we need to apply the product rule: \( (uv)' = u'v + uv' \), where \( u = x^{\pi} \) and \( v = \cosh x \).
- Derivatives:
- \( u' = \pi x^{\pi-1} \)
- \( v' = \sinh x \)
- Applying the product rule:
\[
f'(x) = (\pi x^{\pi-1}) \cosh x + (x^{\pi}) \sinh x
\]
2. **Second Derivative (\( f''(x) \))**:
- Differentiate \( f'(x) \) again using the product rule on each term:
- For \( (\pi x^{\pi-1} \cosh x) \):
- Apply product rule:
\[
\left(\pi x^{\pi-1}\right)' = \pi(\pi-1)x^{\pi-2}
\]
\[
( \pi(\pi-1)x^{\pi-2} ) \cosh x + (\pi x^{\pi-1}) \sinh x
\]
- For \( (x^{\pi} \sinh x) \):
- Apply product rule:
\[
(x^{\pi})' = \pi x^{\pi-1}
\]
\[
(\pi x^{\pi-1}) \sinh x + (x^{\pi}) \cosh x
\]
- Combine terms:
\[
f''(x) = [\pi(\pi-1)x^{\pi-2} \cosh x + \pi x^{\pi-1} \sinh x] + [\pi x^{\pi-1} \sinh x
Expert Solution
Step 1
Given:-
f(x) =xπ coshx
To find:-
f''(x)
Step by step
Solved in 3 steps with 2 images
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