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 \(\frac{dy}{dx}\).
\[ y = \int_{\sqrt{x}}^{\pi/2} \cos(t^2) \, dt \]
**Solution:**
To find \(\frac{dy}{dx}\), use the Fundamental Theorem of Calculus. The theorem states that if an integral has a variable as a limit, the derivative with respect to that variable can be found using:
\[
\frac{dy}{dx} = \frac{d}{dx} \left( \int_{\sqrt{x}}^{\pi/2} \cos(t^2) \, dt \right) = -\cos((\sqrt{x})^2) \cdot \frac{d}{dx}(\sqrt{x})
\]
The derivative of \(\sqrt{x}\) with respect to \(x\) is \(\frac{1}{2\sqrt{x}}\).
Therefore,
\[
\frac{dy}{dx} = -\cos(x) \times \frac{1}{2\sqrt{x}}
\]
\[
\frac{dy}{dx} = -\frac{\cos(x)}{2\sqrt{x}}
\]
The box in the image is a placeholder for the final answer, which is:
\[
\frac{dy}{dx} = -\frac{\cos(x)}{2\sqrt{x}}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93ade889-6aa6-440d-b6d3-23ce5a98ca6e%2Fe4b6dcc4-26bb-45e2-b764-3a99eb387350%2F4yfo0kf_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find \(\frac{dy}{dx}\).
\[ y = \int_{\sqrt{x}}^{\pi/2} \cos(t^2) \, dt \]
**Solution:**
To find \(\frac{dy}{dx}\), use the Fundamental Theorem of Calculus. The theorem states that if an integral has a variable as a limit, the derivative with respect to that variable can be found using:
\[
\frac{dy}{dx} = \frac{d}{dx} \left( \int_{\sqrt{x}}^{\pi/2} \cos(t^2) \, dt \right) = -\cos((\sqrt{x})^2) \cdot \frac{d}{dx}(\sqrt{x})
\]
The derivative of \(\sqrt{x}\) with respect to \(x\) is \(\frac{1}{2\sqrt{x}}\).
Therefore,
\[
\frac{dy}{dx} = -\cos(x) \times \frac{1}{2\sqrt{x}}
\]
\[
\frac{dy}{dx} = -\frac{\cos(x)}{2\sqrt{x}}
\]
The box in the image is a placeholder for the final answer, which is:
\[
\frac{dy}{dx} = -\frac{\cos(x)}{2\sqrt{x}}
\]
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