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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Compute the derivatives
![Below are a series of differential equations, each followed by an expression to differentiate with respect to \( x \). These problems aim to provide practice in differentiation techniques:
(a) \( \dfrac{d}{dx} \left[ \ln(4) - 2e^{\sqrt{3}} \right] \)
(b) \( \dfrac{d}{dx} \left[ x^4 e^{3x} - \dfrac{1}{2} \ln(x^2) \right] \)
(c) \( \dfrac{d}{dx} \left[ \sec^6(12x) + \tan^{-1}(2x) \right] \)
(d) \( \dfrac{d}{dx} \left[ \dfrac{\tan(2x)}{e^{2x} - 13} \right] \)
---
For educators: These problems are designed to test various differentiation rules including the product rule, chain rule, and the derivatives of transcendental functions such as logarithms, exponential functions, and trigonometric functions. Here's a brief guide on how to approach each:
1. **Problem (a)**:
- The natural logarithm function and exponential function are constants with respect to \( x \).
- This should lead to recognizing that the derivative of a constant is zero.
2. **Problem (b)**:
- Apply the product rule for \( x^4 e^{3x} \).
- Use the properties of logarithms and their derivatives for \( \dfrac{1}{2} \ln(x^2) \).
3. **Problem (c)**:
- Implement the chain rule for \( \sec^6(12x) \).
- The derivative of the inverse tangent function, \( \tan^{-1}(2x) \), also requires the chain rule.
4. **Problem (d)**:
- Utilize the quotient rule for the fraction \( \dfrac{\tan(2x)}{e^{2x} - 13} \).
- Apply chain rule within the quotient rule as needed.
Encourage students to become comfortable with these foundational techniques as they are crucial for more advanced calculus topics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7722005-635d-4232-9de7-b0c3c13aa61f%2Fcad51816-8fc2-4304-832d-626c3e7f1519%2Fytvdc6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Below are a series of differential equations, each followed by an expression to differentiate with respect to \( x \). These problems aim to provide practice in differentiation techniques:
(a) \( \dfrac{d}{dx} \left[ \ln(4) - 2e^{\sqrt{3}} \right] \)
(b) \( \dfrac{d}{dx} \left[ x^4 e^{3x} - \dfrac{1}{2} \ln(x^2) \right] \)
(c) \( \dfrac{d}{dx} \left[ \sec^6(12x) + \tan^{-1}(2x) \right] \)
(d) \( \dfrac{d}{dx} \left[ \dfrac{\tan(2x)}{e^{2x} - 13} \right] \)
---
For educators: These problems are designed to test various differentiation rules including the product rule, chain rule, and the derivatives of transcendental functions such as logarithms, exponential functions, and trigonometric functions. Here's a brief guide on how to approach each:
1. **Problem (a)**:
- The natural logarithm function and exponential function are constants with respect to \( x \).
- This should lead to recognizing that the derivative of a constant is zero.
2. **Problem (b)**:
- Apply the product rule for \( x^4 e^{3x} \).
- Use the properties of logarithms and their derivatives for \( \dfrac{1}{2} \ln(x^2) \).
3. **Problem (c)**:
- Implement the chain rule for \( \sec^6(12x) \).
- The derivative of the inverse tangent function, \( \tan^{-1}(2x) \), also requires the chain rule.
4. **Problem (d)**:
- Utilize the quotient rule for the fraction \( \dfrac{\tan(2x)}{e^{2x} - 13} \).
- Apply chain rule within the quotient rule as needed.
Encourage students to become comfortable with these foundational techniques as they are crucial for more advanced calculus topics.
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