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
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![**Evaluate**
\[ \frac{d}{dx} e^{6x^2 + 8x} = \]
The task is to differentiate the given exponential function with respect to \( x \). The expression inside the exponential function is \( 6x^2 + 8x \).
To find the derivative, apply the chain rule:
1. The derivative of \( e^u \) with respect to \( u \) is \( e^u \).
2. The derivative of \( u = 6x^2 + 8x \) with respect to \( x \) is \( 12x + 8 \).
The result is:
\[ \frac{d}{dx} e^{6x^2 + 8x} = e^{6x^2 + 8x} \cdot (12x + 8) \]
Hence, the derivative is:
\[ e^{6x^2 + 8x} \cdot (12x + 8) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6acc7099-89c3-4cde-aa7d-6eca9bf4997e%2Fe942cdd1-fbf5-4eb7-b9bd-9c44e66ab0c7%2Fu7t4jxi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluate**
\[ \frac{d}{dx} e^{6x^2 + 8x} = \]
The task is to differentiate the given exponential function with respect to \( x \). The expression inside the exponential function is \( 6x^2 + 8x \).
To find the derivative, apply the chain rule:
1. The derivative of \( e^u \) with respect to \( u \) is \( e^u \).
2. The derivative of \( u = 6x^2 + 8x \) with respect to \( x \) is \( 12x + 8 \).
The result is:
\[ \frac{d}{dx} e^{6x^2 + 8x} = e^{6x^2 + 8x} \cdot (12x + 8) \]
Hence, the derivative is:
\[ e^{6x^2 + 8x} \cdot (12x + 8) \]
![Suppose that
\[ y = (3x^2 + 4x + 2)^{1/5}. \]
Find \(\frac{dy}{dx}\).
\[
\frac{dy}{dx} = \text{(input box)}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6acc7099-89c3-4cde-aa7d-6eca9bf4997e%2Fe942cdd1-fbf5-4eb7-b9bd-9c44e66ab0c7%2Folostrw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose that
\[ y = (3x^2 + 4x + 2)^{1/5}. \]
Find \(\frac{dy}{dx}\).
\[
\frac{dy}{dx} = \text{(input box)}
\]
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