Q 92 Use implicit differentiation and... [FREE SOLUTION]

Chapter 2: Q 92 (page 212)

Use implicit differentiation and the power rule for integer powers (not the general power rule) to prove that

Short Answer

Expert verified

$y = x^{2 / 3} 鈬� y^{3} = x^{2} 鈬� \frac{d}{d x} y^{3} = \frac{d}{d x} x^{2} 鈬� 3 y^{2} \frac{d y}{d x} = 2 x 鈬� \frac{d y}{d x} = \frac{2 x}{3 y^{2}} 鈬� \frac{d y}{d x} = \frac{2}{3} x y^{- 2} 鈬� 鈬� \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x {(x^{2 / 3})}^{- 2} 鈬� \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x (x^{- 4 / 3}) 鈬� \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x^{1 - 4 / 3} 鈬� \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x^{- 1 / 3}$

Hence proved.

Step by step solution

Step 1. Given Information

We have given that :-

$\frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x^{- 1 / 3}$
We have to prove this derivative by using implicit differentiation and power rule for integer powers.

Step 2. Prove that ddx(x2/3)=23x-1/3

We have to find the derivative of $x^{2 / 3}$ .

Let :-

$y = x^{2 / 3}$ .

We can write it as :-

role="math" localid="1648652545614" $y^{3} = x^{2}$ .

Then by using implicit differentiation :-

role="math" localid="1648652570552" $\frac{d}{d x} y^{3} = \frac{d}{d x} x^{2}$

Then by using chain rule and power rule for integer powers, we have :-

role="math" localid="1648652595546" $3 y^{2} \frac{d y}{d x} = 2 x 鈬� \frac{d y}{d x} = \frac{2 x}{3 y^{2}} 鈬� \frac{d y}{d x} = \frac{2}{3} x y^{- 2}$

Put the value $y = x^{2 / 3}$ , then we have :-

role="math" localid="1648652616933" $鈬� \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x {(x^{2 / 3})}^{- 2} 鈬� \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x (x^{- 4 / 3}) 鈬� \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x^{1 - 4 / 3} 鈬� \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x^{- 1 / 3}$

This is the required value.

Hence proved.

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Use implicit differentiation and the power rule for integer powers (not the general power rule) to prove that

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Prove that ddx(x2/3)=23x-1/3

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Theoretical and Mathematical Physics

Applied Mathematics

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.