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: Differentiation
**Problem Statement:**
Given the function:
\[ f(x) = \frac{-5x^5 + 8x^4 + 7x^3}{x^4} \]
find \( f'(x) \).
**Step-by-Step Solution:**
1. **Simplify the function:**
First, simplify the given function by dividing each term in the numerator by the denominator.
\[ f(x) = \frac{-5x^5}{x^4} + \frac{8x^4}{x^4} + \frac{7x^3}{x^4} \]
2. **Simplify each term:**
Simplify each fraction:
\[
f(x) = -5x^{5-4} + 8x^{4-4} + 7x^{3-4}
\]
This simplifies to:
\[
f(x) = -5x + 8 + 7x^{-1}
\]
3. **Differentiate the simplified function:**
Apply the power rule to differentiate each term:
- The derivative of \(-5x\) is \(-5\).
- The derivative of \(8\) is \(0\).
- The derivative of \(7x^{-1}\) is \(7 \cdot (-1)x^{-1-1} = -7x^{-2}\).
Therefore:
\[
f'(x) = -5 + 0 - 7x^{-2}
\]
Simplifying, we get:
\[
f'(x) = -5 - \frac{7}{x^2}
\]
**Final Answer:**
\[ f'(x) = -5 - \frac{7}{x^2} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f6b4032-4723-40f9-bafe-2ad74429fe12%2F94872da7-37cc-4b80-8cf7-0378b9b753b5%2Fbbkj8k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus Problem: Differentiation
**Problem Statement:**
Given the function:
\[ f(x) = \frac{-5x^5 + 8x^4 + 7x^3}{x^4} \]
find \( f'(x) \).
**Step-by-Step Solution:**
1. **Simplify the function:**
First, simplify the given function by dividing each term in the numerator by the denominator.
\[ f(x) = \frac{-5x^5}{x^4} + \frac{8x^4}{x^4} + \frac{7x^3}{x^4} \]
2. **Simplify each term:**
Simplify each fraction:
\[
f(x) = -5x^{5-4} + 8x^{4-4} + 7x^{3-4}
\]
This simplifies to:
\[
f(x) = -5x + 8 + 7x^{-1}
\]
3. **Differentiate the simplified function:**
Apply the power rule to differentiate each term:
- The derivative of \(-5x\) is \(-5\).
- The derivative of \(8\) is \(0\).
- The derivative of \(7x^{-1}\) is \(7 \cdot (-1)x^{-1-1} = -7x^{-2}\).
Therefore:
\[
f'(x) = -5 + 0 - 7x^{-2}
\]
Simplifying, we get:
\[
f'(x) = -5 - \frac{7}{x^2}
\]
**Final Answer:**
\[ f'(x) = -5 - \frac{7}{x^2} \]
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