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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![Below is the transcription of the given image of a mathematical expression as it would appear on an educational website:
---
### Mathematical Expression for Differentiation
Consider the expression:
\[ \frac{d}{dx} \left( F(x^4) \right) = \frac{d}{dx} \left( \int_{13}^{x^4} e^{-t^2} \, dt \right) \]
In this problem, we are asked to differentiate the function \( F \), which is defined in terms of the integral of \( e^{-t^2} \) with respect to \( t \), where the upper limit of the integral is \( x^4 \).
The expression involves the use of the Fundamental Theorem of Calculus and the Chain Rule. The goal is to find the derivative of the given function with respect to \( x \).
\[\boxed{\phantom{answer box}}\]
---
In the image, there is a text box intended for the answer. On the educational website, you would be expected to input the derived result into this box.
Explanation of Steps:
1. **Fundamental Theorem of Calculus**: If \( F(x) = \int_{a}^{g(x)} f(t) \, dt \), then \( F'(x) = f(g(x)) \cdot g'(x) \).
2. **Applying the Derivative**: Differentiate with respect to \( x \) considering the chain rule, as the upper limit of the integral is \( x^4 \).
3. **Simplify the Result**: Substitute and simplify the expression to obtain the final derivative.
The specific steps can be covered in subsequent detailed sections or lectures.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5e65866-2b52-486f-8280-aebe41369013%2Fca3bf18a-ff54-46d8-b28f-620e2a06e977%2F4e40ta_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Below is the transcription of the given image of a mathematical expression as it would appear on an educational website:
---
### Mathematical Expression for Differentiation
Consider the expression:
\[ \frac{d}{dx} \left( F(x^4) \right) = \frac{d}{dx} \left( \int_{13}^{x^4} e^{-t^2} \, dt \right) \]
In this problem, we are asked to differentiate the function \( F \), which is defined in terms of the integral of \( e^{-t^2} \) with respect to \( t \), where the upper limit of the integral is \( x^4 \).
The expression involves the use of the Fundamental Theorem of Calculus and the Chain Rule. The goal is to find the derivative of the given function with respect to \( x \).
\[\boxed{\phantom{answer box}}\]
---
In the image, there is a text box intended for the answer. On the educational website, you would be expected to input the derived result into this box.
Explanation of Steps:
1. **Fundamental Theorem of Calculus**: If \( F(x) = \int_{a}^{g(x)} f(t) \, dt \), then \( F'(x) = f(g(x)) \cdot g'(x) \).
2. **Applying the Derivative**: Differentiate with respect to \( x \) considering the chain rule, as the upper limit of the integral is \( x^4 \).
3. **Simplify the Result**: Substitute and simplify the expression to obtain the final derivative.
The specific steps can be covered in subsequent detailed sections or lectures.
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