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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![**Calculus Problem: Derivative Calculation Exercise**
**Problem Statement:**
Find the derivative of \( f(x) = \frac{5}{2} x^5 \).
---
**Solution Approach:**
To solve this problem, we will apply the basic rules of differentiation. Specifically, we will use the Power Rule for differentiation, which states:
\[ \frac{d}{dx} [x^n] = n x^{n-1} \]
Keeping the coefficient \( \frac{5}{2} \) in mind, we proceed as follows:
1. Identify the function for differentiation: \( f(x) = \frac{5}{2} x^5 \).
2. Apply the Power Rule to \( x^5 \):
\[ \frac{d}{dx} [x^5] = 5x^4 \]
3. Multiply by the constant coefficient \( \frac{5}{2} \):
\[ \frac{d}{dx} \left( \frac{5}{2} x^5 \right) = \frac{5}{2} \cdot 5 x^4 = \frac{25}{2} x^4 \]
Thus, the derivative of the function \( f(x) = \frac{5}{2} x^5 \) is:
\[ f'(x) = \frac{25}{2} x^4 \]
This concludes the solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F75758824-d9ac-45ff-8935-fa266cdc3747%2F61c7c5ae-c08a-42ac-a6fb-bc75d438329c%2Fcmatc9.png&w=3840&q=75)
Transcribed Image Text:**Calculus Problem: Derivative Calculation Exercise**
**Problem Statement:**
Find the derivative of \( f(x) = \frac{5}{2} x^5 \).
---
**Solution Approach:**
To solve this problem, we will apply the basic rules of differentiation. Specifically, we will use the Power Rule for differentiation, which states:
\[ \frac{d}{dx} [x^n] = n x^{n-1} \]
Keeping the coefficient \( \frac{5}{2} \) in mind, we proceed as follows:
1. Identify the function for differentiation: \( f(x) = \frac{5}{2} x^5 \).
2. Apply the Power Rule to \( x^5 \):
\[ \frac{d}{dx} [x^5] = 5x^4 \]
3. Multiply by the constant coefficient \( \frac{5}{2} \):
\[ \frac{d}{dx} \left( \frac{5}{2} x^5 \right) = \frac{5}{2} \cdot 5 x^4 = \frac{25}{2} x^4 \]
Thus, the derivative of the function \( f(x) = \frac{5}{2} x^5 \) is:
\[ f'(x) = \frac{25}{2} x^4 \]
This concludes the solution.
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