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...
Related questions
Question
Instructions say to find the first derivative.
![The image shows a mathematical expression involving differentiation and integration. The expression is:
\[
\frac{d}{dx} \int_{0}^{x^2} \sin(t) \, dt
\]
This expression represents the derivative with respect to \( x \) of the integral of \( \sin(t) \) with respect to \( t \) from 0 to \( x^2 \).
### Key Components:
1. **Integral**:
\[
\int_{0}^{x^2} \sin(t) \, dt
\]
This is a definite integral where the function \( \sin(t) \) is integrated with respect to \( t \) from 0 to \( x^2 \).
2. **Derivative**:
\[
\frac{d}{dx}
\]
This indicates taking the derivative of the integral with respect to \( x \).
### Explanation:
- The inner part:
\[
\int_{0}^{x^2} \sin(t) \, dt
\]
results in a value that depends on \( x^2 \), and thus on \( x \).
- The derivative:
\[
\frac{d}{dx}
\]
acts on this value. Essentially, this problem involves the method of differentiation under the integral sign, often referred to as the Leibniz rule or the Fundamental Theorem of Calculus when dealing with variable limits of integration.
### Step-by-Step Solution:
1. **Substitute and Differentiate**: According to the Leibniz rule, if we have \( \int_{a(x)}^{b(x)} f(t) \, dt \), then:
\[
\frac{d}{dx} \int_{a(x)}^{b(x)} f(t) \, dt = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)
\]
2. **Apply to our integral**:
Here, \( a(x) = 0 \) and \( b(x) = x^2 \), so:
\[
\frac{d}{dx} \int_{0}^{x^2} \sin(t) \, dt = \sin(x^2) \cdot \frac{d}{dx} (x^2) - \sin(0) \cdot](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3e7eb8b-0b75-4c6c-a4cc-30a54bf299fc%2F70fe5a8a-25fb-47cd-ba41-f44c8bf6c250%2Fmefqqf_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows a mathematical expression involving differentiation and integration. The expression is:
\[
\frac{d}{dx} \int_{0}^{x^2} \sin(t) \, dt
\]
This expression represents the derivative with respect to \( x \) of the integral of \( \sin(t) \) with respect to \( t \) from 0 to \( x^2 \).
### Key Components:
1. **Integral**:
\[
\int_{0}^{x^2} \sin(t) \, dt
\]
This is a definite integral where the function \( \sin(t) \) is integrated with respect to \( t \) from 0 to \( x^2 \).
2. **Derivative**:
\[
\frac{d}{dx}
\]
This indicates taking the derivative of the integral with respect to \( x \).
### Explanation:
- The inner part:
\[
\int_{0}^{x^2} \sin(t) \, dt
\]
results in a value that depends on \( x^2 \), and thus on \( x \).
- The derivative:
\[
\frac{d}{dx}
\]
acts on this value. Essentially, this problem involves the method of differentiation under the integral sign, often referred to as the Leibniz rule or the Fundamental Theorem of Calculus when dealing with variable limits of integration.
### Step-by-Step Solution:
1. **Substitute and Differentiate**: According to the Leibniz rule, if we have \( \int_{a(x)}^{b(x)} f(t) \, dt \), then:
\[
\frac{d}{dx} \int_{a(x)}^{b(x)} f(t) \, dt = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)
\]
2. **Apply to our integral**:
Here, \( a(x) = 0 \) and \( b(x) = x^2 \), so:
\[
\frac{d}{dx} \int_{0}^{x^2} \sin(t) \, dt = \sin(x^2) \cdot \frac{d}{dx} (x^2) - \sin(0) \cdot
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning