/*! 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} Q. 88 Prove the difference rule in two... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Q. 88 (page 199)

Prove the difference rule in two ways

a) using definition of the derivative

b) using sum and constant multiple rules

Short Answer

Expert verified

We prove the difference rules using the given rules

Step by step solution

01

Given information

We are given two function f anf g

02

Prove the difference rule using the definition of the derivative

We get,

limh鈫�0(f-g)(x+h)-(f-g)(x)hlimh鈫�0f(x+h)-g(x+h)-f(x)+g(x)hlimh鈫�0f(x+h)-f(x)h-limh鈫�0g(x+h)-g(x)h=f'(x)-g'(x)Henceproved

03

Proof using sum and constant times rule

We have,

Wearegiven(f-g)'(x)

Which can also be written as

(f+(-g))'(x)f'(x)+(-g'(x))(Sumrule)f'(x)-g'(x)(constantrule)

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

Use (a) the h鈫�0definition of the derivative and then

(b) the z鈫�cdefinition of the derivative to find f'(c) for each function f and value x=c in Exercises 23鈥�38.

26.f(x)=1x2,x=2

If Katie walked at 3miles per hour for 20minutes and then sprinted at 10miles an hour for 8minutes, how fast would Dave have to walk or run to go the same distance as Katie did at the same time while moving at a constant speed? Sketch a graph of Katie鈥檚 position over time and a graph of Dave鈥檚 position over time on the same set of axes.

A tomato plant given xounces of fertilizer will successfully bear T(x)pounds of tomatoes in a growing season.

(a) In real-world terms, what does T(5)represent and what are its units? What does T'(5)represent and what are its units?

(b) A study has shown that this fertilizer encourages tomato production when less than 20ounces are used, but inhibits production when more than 20ounces are used. When is T(x)positive and when is T(x)negative? When is T'(x)positive and when is T'(x)negative?

In Exercises 69-80, determine whether or not fis continuous and/or differentiable at the given value of x. If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative.

f(x)=1x,x=0

Use the definition of the derivative to find ffor each function fin Exercises 39-54

f(x)=1x2-1
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

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.