Recall that the derivative of y = f(x) at any point x is:
2. The Differentials
dy = f '(a) dx.
At a particular point x = a in dom( f
), the function dy = f
is called the differential of
at x = a. Each point
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
be differentiable at x = a. The differential of f at a is the function defined by dy =
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
3. Differentials Vs Infinitesimals
= 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
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
Remember, there are 2
interpretations for the set of the quantities dx
and dy: differentials and infinitesimals.
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.
4. dy/dx As A Normal Fraction
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:
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) = x2 – 3x + 1 at x = 5.
f(x) = x2 – 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/(x2 – 2)3 at x = 0.
3. Find the differential of the function y = x/(x + 1).
4. Find the differential of the function s = etan 3t , Use the formulas (d/dx) ex = ex and (d/dx) tan x = sec2 x.
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)