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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![**Finding the Tangent Line to a Function**
**Problem Statement:**
Find an equation of the tangent line to the graph of the function at the given point.
Function: \( y = 2 \arcsin x \)
Point: \( \left( \frac{1}{2}, \frac{\pi}{3} \right) \)
---
**Step 1: Differentiating the Function**
First, differentiate both sides of the given function \( y = 2 \arcsin x \) with respect to \( x \). Write the differential equation as follows:
\[
y' = \frac{d}{dx}(2 \arcsin x)
\]
Recall the derivative rule for the arcsine function:
\[
\frac{d}{dx}[\arcsin u] = \frac{u'}{\sqrt{1-u^2}}
\]
Rewriting the derivative in the context of our function:
\[
y' = \frac{2}{\sqrt{1-x^2}}
\]
This is the derivative of \( y = 2 \arcsin x \) with respect to \( x \). This derivative will be used to find the slope of the tangent line at the given point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabeb9516-f9a9-404e-800e-8596b81a7ba1%2F5edc718b-526c-450d-abf2-c4e2842d88ea%2Fqsn4db5_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Tangent Line to a Function**
**Problem Statement:**
Find an equation of the tangent line to the graph of the function at the given point.
Function: \( y = 2 \arcsin x \)
Point: \( \left( \frac{1}{2}, \frac{\pi}{3} \right) \)
---
**Step 1: Differentiating the Function**
First, differentiate both sides of the given function \( y = 2 \arcsin x \) with respect to \( x \). Write the differential equation as follows:
\[
y' = \frac{d}{dx}(2 \arcsin x)
\]
Recall the derivative rule for the arcsine function:
\[
\frac{d}{dx}[\arcsin u] = \frac{u'}{\sqrt{1-u^2}}
\]
Rewriting the derivative in the context of our function:
\[
y' = \frac{2}{\sqrt{1-x^2}}
\]
This is the derivative of \( y = 2 \arcsin x \) with respect to \( x \). This derivative will be used to find the slope of the tangent line at the given point.
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