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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![## Differentiation Problem
**Problem Statement:**
Differentiate the function.
\[ y = (3x - 1)^4 \cdot (4 - x^5)^3 \]
**Solution:**
To differentiate the given function, we need to apply the chain rule and the product rule.
1. **Identify components of the function:**
- First Part: \( (3x - 1)^4 \)
- Second Part: \( (4 - x^5)^3 \)
2. **Differentiating each part:**
- Let \( u = (3x - 1)^4 \)
- Let \( v = (4 - x^5)^3 \)
We need to find \( \frac{d}{dx} [ u \cdot v ] \). Using the product rule:
\[
\frac{dy}{dx} = u' \cdot v + u \cdot v'
\]
3. **Finding \( u' \) and \( v' \):**
- For \( u = (3x - 1)^4 \):
\[
u' = 4(3x - 1)^3 \cdot 3 = 12(3x - 1)^3
\]
- For \( v = (4 - x^5)^3 \):
\[
v' = 3(4 - x^5)^2 \cdot (-5x^4) = -15x^4 (4 - x^5)^2
\]
4. **Substituting back into the product rule formula:**
\[
\frac{dy}{dx} = (12(3x - 1)^3) \cdot (4 - x^5)^3 + (3x - 1)^4 \cdot (-15x^4 (4 - x^5)^2)
\]
Thus, this derivative provides the solved form for the given function.
Please enter your calculated derivative in the box provided.
\[
\frac{dy}{dx} = \boxed{}
\]
In this step-by-step process, you can differentiate the given compound function by carefully applying differentiation rules.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F68ad5c49-9c00-40e0-b55c-449508bbd9d9%2F652ad23b-57f4-40a5-afa5-c337d8336789%2Frjc6x1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Differentiation Problem
**Problem Statement:**
Differentiate the function.
\[ y = (3x - 1)^4 \cdot (4 - x^5)^3 \]
**Solution:**
To differentiate the given function, we need to apply the chain rule and the product rule.
1. **Identify components of the function:**
- First Part: \( (3x - 1)^4 \)
- Second Part: \( (4 - x^5)^3 \)
2. **Differentiating each part:**
- Let \( u = (3x - 1)^4 \)
- Let \( v = (4 - x^5)^3 \)
We need to find \( \frac{d}{dx} [ u \cdot v ] \). Using the product rule:
\[
\frac{dy}{dx} = u' \cdot v + u \cdot v'
\]
3. **Finding \( u' \) and \( v' \):**
- For \( u = (3x - 1)^4 \):
\[
u' = 4(3x - 1)^3 \cdot 3 = 12(3x - 1)^3
\]
- For \( v = (4 - x^5)^3 \):
\[
v' = 3(4 - x^5)^2 \cdot (-5x^4) = -15x^4 (4 - x^5)^2
\]
4. **Substituting back into the product rule formula:**
\[
\frac{dy}{dx} = (12(3x - 1)^3) \cdot (4 - x^5)^3 + (3x - 1)^4 \cdot (-15x^4 (4 - x^5)^2)
\]
Thus, this derivative provides the solved form for the given function.
Please enter your calculated derivative in the box provided.
\[
\frac{dy}{dx} = \boxed{}
\]
In this step-by-step process, you can differentiate the given compound function by carefully applying differentiation rules.
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