   Chapter 3.10, Problem 41E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Establish the following rules for working with differentials (where c denotes a constant and u and v are functions of x).(a) dc = 0(b) d(cu) = c du(c) d(u + v) = du + dv(d) d(uv) = u dv + v du(e) d ( u v ) = v   d u − u   d v v 2 (f) d(xn) = nxn–1 dx

(a)

To determine

To establish: The rule dc=0 for working with differential, where c is a constant.

Explanation

Consider the function y=c.

The differential is dy=f(x)dx.

The derivative of the function f(x)=c is f(x)=0

(b)

To determine

To establish: The rules d(cu)=cdu  for working with differential, where c is a constant and u is a function of x.

(c)

To determine

To establish: The rules d(u+v)=du+dv  for working with differential.

(d)

To determine

To establish: The rules d(uv)=udv+vdu  for working with differential.

(e)

To determine

To establish: The rules d(uv)=vduudvv2  for working with differential.

(f)

To determine

To establish: The rules d(xn)=nxn1dx for working with differential.

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