/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Find the derivative of the funct... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 39

Find the derivative of the function. \(f(x)=2 \arcsin (x-1)\)

Short Answer

Expert verified
The derivative of the function \(f(x) = 2\arcsin(x - 1)\) is \(f'(x) = \frac{2}{\sqrt{2x - x^2}}\).

Step by step solution

01

Identify the outer function and the inner function

In the given function \(f(x) = 2\arcsin(x-1)\), the outer function is \(2\arcsin(u)\) and the inner function is \(u = x - 1\). The chain rule states that the derivative of a composite function is the derivative of the outer function, evaluated at the inner function, times the derivative of the inner function.
02

Find the derivatives of the outer function and the inner function

The derivative of the outer function \(2\arcsin(u)\) with respect to \(u\) is \(\frac{2}{\sqrt{1-u^2}}\). The derivative of the inner function \(x - 1\) with respect to \(x\) is 1.
03

Apply the chain rule

The chain rule states that the derivative of the composite function \(f(x)\) with respect to \(x\) is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Therefore, \(f'(x) = \frac{2}{\sqrt{1-(x - 1)^2}} \cdot 1\).
04

Simplify the derivative

The expression inside the square root \(1-(x - 1)^2 = 2x - x^2\). Consequently, the derivative simplifies to \(f'(x) = \frac{2}{\sqrt{2x - x^2}}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the derivative of the function. \(f(x)=\arctan \sqrt{x}\)

Use implicit differentiation to find an equation of the tangent line to the graph of the equation at the given point. \(\arcsin x+\arcsin y=\frac{\pi}{2}, \quad\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)

(a) Use a graphing utility to evaluate arcsin (arcsin 0.5) and \(\arcsin (\arcsin 1) .\) (b) Let \(f(x)=\arcsin (\arcsin x)\) Find the values of \(x\) in the interval \(-1 \leq x \leq 1\) such that \(f(x)\) is a real number.

Find the derivative of the function. \(y=x \arcsin x+\sqrt{1-x^{2}}\)

Find the derivative of the function. \(f(x)=\operatorname{arcsec} 2 x\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.