Calculus Of One Real Variable
– By Pheng Kim Ving

4.3 
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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.
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 = iii. Let
the function y = f(x) be differentiable at all points of dom( f ).The differential
of f is the operation that

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 nonstandard
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.
1. Find the differential of y = f(x) = x^{2} – 3x + 1 at x = 5.
f(x) = x^{2} – 3x + 1,
f '(x) = 2x – 3,
f '(5) = 2(5) – 3 = 7,
dy = f '(5) dx = 7 dx.
2. Find the differential of y = (x + 1)^{2}/(x^{2} – 2)^{3} at x = 0.
3. Find the differential of the function y = x/(x + 1).
Solution
4. Find the differential of the function s = e^{tan}^{ 3}^{t} , Use the formulas (d/dx) e^{x} = e^{x} and (d/dx) tan x = sec^{2} x.
Solution
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:
= f '(a) – f '(a)
= 0.
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