Calculus Of One Real Variable By Pheng Kim Ving
Chapter 4: More On The Derivative Section 4.3: The Differentials

 

4.3
The Differentials

 

 

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1. Infinitesimals

 

Recall that the derivative of y = f(x) at any point x is:

 

 

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2. The Differentials

 

 

Fig. 2.1

 

dy = f '(a) dx.

 

At a particular point x = a in dom( f ), the function dy = f '(a) dx is called the differential of f at x = a. Each point of
dom( f ) generates the differential of f at that point. The set of all points of dom( f ) generates the set of all the
differentials of f at those points. This set of differentials is called the differential of f. Formally the differential of f is
defined as the operation that associates or maps dom( f ) to this set of differentials.

 

Definitions 2.1

 

i. The differential of x is a change in x, denoted dx.

 

ii. Let the function y = f(x) be differentiable at x = a. The differential of f at a is the function defined by dy =
f '(a) dx, where the differential dx of x is the independent variable and dy the dependent variable.

 

iii. Let the function y = f(x) be differentiable at all points of dom( f ).The differential of f is the operation that
associates to each point x in dom ( f ) the differential dy = f '(x) dx of f at that point. As dy = f '(x) dx we have
dy/dx = f '(x). So:

 

 

 

Remarks 2.1

 

 

also. This has led us to denote differentials by the letter d, like dx or dy or dt, the same as the notation for
infinitesimals.

 

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3. Differentials Vs Infinitesimals

 

Let y = f(x). The differential dy of f at an arbitrary point x in dom ( f ) is dy = f '(x) dx, where dx is the differential of
x. So dy/dx = f '(x). That is, the ratio dy/dx, where dy and dx are differentials, is the derivative of y with respect to x.
This is in accordance with the Leibniz notation dy/dx of the derivative of y with respect to x, where dy and dx are
infinitesimals.

 

When Leibniz introduced the concept of infinitesimals, he was roundly criticized for inventing such infinitesimals out of thin
air and attempting to do mathematics with them. It's only during the last 60 years or so that mathematicians have put his
view on a firm mathematical foundation by showing that it's possible to extend the set of real numbers to include such
infinitesimals.

 

Remember, there are 2 interpretations for the set of the quantities dx and dy: differentials and infinitesimals. Calculus
with dx and dy interpreted as differentials is called standard analysis. Calculus with dx and dy interpreted as
infinitesimals and the set of real numbers extended to include them is called non-standard analysis.

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4. dy/dx As A Normal Fraction

 

If dx and dy are interpreted as differentials, the quotient dy/dx is a normal fraction because dx and dy are each a real
number. If dx and dy are interpreted as infinitesimals, the quotient dy/dx is a normal fraction because dx and dy are
each a real number in the extended set of real numbers. Whether dx and dy are interpreted as differentials or as
infinitesimals, the ratio:

 

 

where the du's appear to cancel out, like in normal fractions. We'll also see times and again later on that the notation
dy/dx indeed behaves like a normal fraction.

 

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Problems & Solutions

 

1. Find the differential of y = f(x) = x2 3x + 1 at x = 5.

 

Solution

 

f(x) = x2 3x + 1,

f '(x) = 2x 3,

f '(5) = 2(5) 3 = 7,

dy = f '(5) dx = 7 dx.

 

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2. Find the differential of y = (x + 1)2/(x2 2)3 at x = 0.

 

Solution

 

 

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3. Find the differential of the function y = x/(x + 1).

 

Solution

 

 

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4. Find the differential of the function s = etan 3t , Use the formulas (d/dx) ex = ex and (d/dx) tan x = sec2 x.

 

Solution

 

 

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5. Let y = f(x) be a differentiable function. See the figure below. Let a be a point in dom( f ) and dx a differential of x.

 

We claim that this error E(dx) is small (close to 0) compared with the size of dx if dx itself is sufficiently small.
Prove this claim by proving that:

 

 

 

Solution

 

 

= f '(a) f '(a)

 

= 0.

 

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