/*! 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 33 If \(y=\frac{x}{x+2}\), prove th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 33

If \(y=\frac{x}{x+2}\), prove that \(x \frac{d y}{d x}=(1-y) y\).

Short Answer

Expert verified
After applying the quotient rule to the given function, multiplying by x, and simplifying, it can be confirmed that \(x \frac{d y}{d x} = (1-y)y\), thus the given equation is proven valid.

Step by step solution

01

Differentiate y with respect to x

Find the derivative \( \frac{d y}{d x} \) of \(y=\frac{x}{x+2}\) using the quotient rule which states that \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \). Here, \( u = x \), \( v = x + 2 \), \(\frac{du}{dx}=1 \), and \(\frac{dv}{dx}=1 \). Apply these values to the quotient rule formula, and simplify to yield \( \frac{d y}{d x} \).
02

Multiply by x

Next, as per the question, multiply \( \frac{d y}{d x} \) by \( x \) to give \( x \frac{d y}{d x} \), and then simplify.
03

Simplify the expression for (1-y)y

Simplify the expression \((1-y)y\), where \( y = \frac{x}{x+2} \). Replace \( y \) with the given expression, \( \frac{x}{x+2} \) and simplify.
04

Compare the right-hand side and left-hand side

Finally, check whether the simplified forms of \( x \frac{d y}{d x} \) and \( (1-y)y \) are equal. If they are equal, the given equation is proven.

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 \(x=a\left(t+\frac{1}{t}\right)\) and \(y=a\left(t-\frac{1}{t}\right)\), then prove that \(\frac{d y}{d x}=\frac{x}{y}\)

If \(y=\sin x \cdot \sin 2 x \cdot \sin 3 x \ldots \sin (2014) x\), find \(\frac{d y}{d x}\)

If \(y=\tan ^{-1}\left\\{\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right\\}\), find \(\frac{d y}{d x}\).

If \(y=A \cos (\log x)+B \sin (\log x)\), then prove that \(x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0\)

If \(y=2 \sin x+3 \cos x\), prove that, \(\frac{d^{2} y}{d x^{2}}+y=0\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Pure Maths

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.