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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![### Problem Statement
Compute the derivative with respect to \( x \) of the expression:
\[
\frac{d}{dx} \left( \frac{1}{e^{2x-1}} - 2x^3 \right).
\]
### Explanation
This problem involves finding the derivative of a function that includes both an exponential and a polynomial term.
1. **Exponential Term**: The first part, \(\frac{1}{e^{2x-1}}\), can be rewritten using the power rule for derivatives and the chain rule. We need to find the derivative of a function in the form of \(e\) to a variable power.
2. **Polynomial Term**: The second part, \(-2x^3\), is a simple polynomial function, and its derivative can be found using the basic rules for differentiating polynomials.
### Steps for Derivation
#### 1. Differentiate the Exponential Term:
- Rewrite \(\frac{1}{e^{2x-1}}\) as \(e^{-(2x-1)}\).
- Use the chain rule: \(\frac{d}{dx} [e^{u(x)}] = e^{u(x)} \cdot u'(x)\).
- Calculate the derivative of the exponent \(- (2x - 1)\) with respect to \(x\).
#### 2. Differentiate the Polynomial Term:
- Use the power rule: \(\frac{d}{dx} [x^n] = nx^{n-1}\).
- Apply this to \(-2x^3\) to find its derivative.
### Solution Box
Provide the final derivative solution here after performing the necessary computations for each term.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe4879982-7bbd-420e-b7e0-63c24befc0ac%2Fe636d9f8-c4ff-42ef-903e-0031a0a4e0cc%2F9mm9uyc_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Compute the derivative with respect to \( x \) of the expression:
\[
\frac{d}{dx} \left( \frac{1}{e^{2x-1}} - 2x^3 \right).
\]
### Explanation
This problem involves finding the derivative of a function that includes both an exponential and a polynomial term.
1. **Exponential Term**: The first part, \(\frac{1}{e^{2x-1}}\), can be rewritten using the power rule for derivatives and the chain rule. We need to find the derivative of a function in the form of \(e\) to a variable power.
2. **Polynomial Term**: The second part, \(-2x^3\), is a simple polynomial function, and its derivative can be found using the basic rules for differentiating polynomials.
### Steps for Derivation
#### 1. Differentiate the Exponential Term:
- Rewrite \(\frac{1}{e^{2x-1}}\) as \(e^{-(2x-1)}\).
- Use the chain rule: \(\frac{d}{dx} [e^{u(x)}] = e^{u(x)} \cdot u'(x)\).
- Calculate the derivative of the exponent \(- (2x - 1)\) with respect to \(x\).
#### 2. Differentiate the Polynomial Term:
- Use the power rule: \(\frac{d}{dx} [x^n] = nx^{n-1}\).
- Apply this to \(-2x^3\) to find its derivative.
### Solution Box
Provide the final derivative solution here after performing the necessary computations for each term.
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