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:**
Find the derivative of the function.
**Given Function:**
\[
g(x) = \int_{3x}^{7x} \frac{2 - 5}{u^2 + 5} \, du
\]
**Hint:**
\[
\int_{3x}^{7x} f(u) \, du = \int_{3x}^{0} f(u) \, du + \int_{0}^{7x} f(u) \, du
\]
**Solution:**
Find \( g'(x) \).
---
### Explanation
In this problem, we are asked to find the derivative of the integral function \( g(x) \). The integral has variable limits of integration from \( 3x \) to \( 7x \) and involves the function \( \frac{2 - 5}{u^2 + 5} \).
The provided hint suggests breaking the integral from \( 3x \) to \( 7x \) into two parts: one from \( 3x \) to \( 0 \), and the other from \( 0 \) to \( 7x \). This technique is useful when applying the Leibniz rule for differentiation under the integral sign, especially with variable limits.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5abad2ef-7d64-4544-b6f7-dd3dfbff74f8%2F2c286e21-97fd-4fbf-a058-01367a6d434f%2Fomov8m_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the derivative of the function.
**Given Function:**
\[
g(x) = \int_{3x}^{7x} \frac{2 - 5}{u^2 + 5} \, du
\]
**Hint:**
\[
\int_{3x}^{7x} f(u) \, du = \int_{3x}^{0} f(u) \, du + \int_{0}^{7x} f(u) \, du
\]
**Solution:**
Find \( g'(x) \).
---
### Explanation
In this problem, we are asked to find the derivative of the integral function \( g(x) \). The integral has variable limits of integration from \( 3x \) to \( 7x \) and involves the function \( \frac{2 - 5}{u^2 + 5} \).
The provided hint suggests breaking the integral from \( 3x \) to \( 7x \) into two parts: one from \( 3x \) to \( 0 \), and the other from \( 0 \) to \( 7x \). This technique is useful when applying the Leibniz rule for differentiation under the integral sign, especially with variable limits.
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