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 Statement:**
2. Find
\[
\frac{d}{dx} \left[ \frac{e^x (x^4 + 1)^x}{x} \right].
\]
**Explanation:**
This problem involves finding the derivative of a function using calculus techniques. The expression inside the derivative is a fraction, where the numerator is the product of exponential and polynomial functions, raised to a power, and the denominator is a linear function of x.
To solve the problem, apply the quotient rule and the product rule. Additionally, you'll need to use the chain rule to handle the composite function within the numerator.
**Steps to Approach:**
1. Identify the functions:
- Let \( u(x) = e^x (x^4 + 1)^x \) (numerator)
- Let \( v(x) = x \) (denominator)
2. Apply the quotient rule:
- The quotient rule states: \(\frac{d}{dx} \left[ \frac{u}{v} \right] = \frac{v \cdot u' - u \cdot v'}{v^2}\)
3. Differentiate \( u(x) \) using the product and chain rules.
4. Substitute the derivatives back into the quotient rule formula to find the complete derivative of the given expression.
**Additional Notes:**
- The problem tests understanding of multiple differentiation rules.
- Simplification of the expression might require factoring and algebraic manipulation.
- It's important to ensure all rules are applied correctly for an accurate result.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf00542e-b5b8-4896-9f9f-a2b46551c658%2Fd2904e6f-67a0-4331-80d9-e8b373117fd1%2F2ywj03c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
2. Find
\[
\frac{d}{dx} \left[ \frac{e^x (x^4 + 1)^x}{x} \right].
\]
**Explanation:**
This problem involves finding the derivative of a function using calculus techniques. The expression inside the derivative is a fraction, where the numerator is the product of exponential and polynomial functions, raised to a power, and the denominator is a linear function of x.
To solve the problem, apply the quotient rule and the product rule. Additionally, you'll need to use the chain rule to handle the composite function within the numerator.
**Steps to Approach:**
1. Identify the functions:
- Let \( u(x) = e^x (x^4 + 1)^x \) (numerator)
- Let \( v(x) = x \) (denominator)
2. Apply the quotient rule:
- The quotient rule states: \(\frac{d}{dx} \left[ \frac{u}{v} \right] = \frac{v \cdot u' - u \cdot v'}{v^2}\)
3. Differentiate \( u(x) \) using the product and chain rules.
4. Substitute the derivatives back into the quotient rule formula to find the complete derivative of the given expression.
**Additional Notes:**
- The problem tests understanding of multiple differentiation rules.
- Simplification of the expression might require factoring and algebraic manipulation.
- It's important to ensure all rules are applied correctly for an accurate result.
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