/*! 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 52 If \(y=\sin ^{-1}\left(x \sqrt{1... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 52

If \(y=\sin ^{-1}\left(x \sqrt{1-x}-\sqrt{x-x^{3}}\right)\), find \(\frac{d y}{d x}\)

Short Answer

Expert verified
The derivative of the given function \(y=\sin ^{-1}\left(x \sqrt{1-x}-\sqrt{x-x^{3}}\right)\) is \(\frac{ dx}{du} = \frac{1}{\sqrt{1 - (x\sqrt{1-x}-\sqrt{x-x^3})^2}} \cdot (\sqrt{1-x} + \frac{-x}{2\sqrt{1-x}} - \frac{1-3x^2}{2\sqrt{x-x^3}})\)

Step by step solution

01

Identify the outer function and inner functions

The outer function here is the inverse sine function, and the inner function is \(u = x \sqrt{1-x} - \sqrt{x-x^{3}}\) . In order to take the derivative of this function, we need to apply the chain rule, which means we need to find the derivative of the outer function and multiply it by the derivative of the inner function.
02

Differentiating the outer function

The derivative of the inverse sine function is given by \(\frac{d}{du}(\sin^{-1}u) = \frac{1}{\sqrt{1 - u^2}}\). So the derivative of \(\sin^{-1}u\) is \( \frac{1}{\sqrt{1 - (x\sqrt{1-x}-\sqrt{x-x^3})^2}}\). Note that we need to square the entire inner function.
03

Differentiate the inner function

Now, we need to differentiate the inner function. This requires the use of the product rule on the first term and the chain rule on the second term. The derivative of the first term \(x\sqrt{1-x}\) is \( \sqrt{1-x}+x \cdot \frac{-1}{2\sqrt{1-x}}\) and the derivative of the second term \(\sqrt{x-x^3}\) is \( \frac{1}{2\sqrt{x-x^3}} \cdot (1-3x^2)\). The combined derivative will be the difference of these two.
04

Apply the chain rule

Now we combine the derivative of the outer function with the derivative of the inner function. Multiplying the two derivatives together and simplifying, we get the final answer.

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

If \(\tan ^{-1}\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)=a\) prove that \(\frac{d y}{d x}=\frac{x(1-\tan a)}{y(1+\tan a)}\).

If \(y=\cos ^{-1}\left(\frac{5 t+12 \sqrt{1-t^{2}}}{13}\right)\) and \(x=\cos ^{-1}\left(\frac{1-t^{2}}{1+t^{2}}\right)\), find \(\frac{d y}{d x}\)

If \(y=\cos ^{-1}\left(\sqrt{\frac{\cos 3 x}{\cos ^{3} x}}\right)\), prove that \(\frac{d y}{d x}=\sqrt{\frac{6}{\cos 2 x+\cos 4 x}}\)

If \(y=\left(C_{1}+C_{2} x\right) \sin x+\left(C_{3}+C_{4} x\right) \cos x\), show that \(\frac{d^{4} y}{d x^{4}}+2 \frac{d^{2} y}{d x^{2}}+y=0\)

Find the value of $$ \lim _{x \rightarrow 0}\left(\frac{1-\cos x \cos 2 x \cos 3 x}{x^{2}}\right) $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.