2. Differentials are undoubtedly one of the most esoteric and least
understood aspects of calculus.
A differential is an infinitessimal change in a variable, which
means the change is smaller in magnitude than any nonzero real
number.
From the perspective of real numbers, the only infinitessimal is 0,
but a change of 0 in a variable is not particularly useful or
exciting.
Are there any more useful infinitessimals out there?
The answer is yes, but we need to look beyond the real numbers
to find them.
Introduction
3. An alternate (but equally valid) way of developing the theory of
calculus is using infinitessimals, instead of limits.
In fact, this is the way that the calculus was initially developed by
Newton and Leibniz (although it was not until many years that
this notion was given a rigorous mathematical base, which is one
of the reasons we develop calculus using limits; the other reason is
that limits have many more applications other than just
differentiation and integration)
Introduction
4. In order to develop calculus in this way, we need to work with the
hyperreal number system, in which there are nonzero
infinitessimals.
In this way we can solve the problem of looking at the rate of
change of a function over an interval as the length of the interval
approaches 0 instead by consider the rate of change of a function
over an interval of infinitessimal length.
Thus, when we write
discussion
5. we are saying that the derivative of f is the ratio of an infinitessimal
change in f over an infinitessimal change in x. We might also write
where the final statement emphasizes explicitly what an
infinitessimal change in f is.
The notation for integration also stems from the notion of
infinitessimals, where we sum up rectangles of infinitessimal width.
We will not go further into this subject, because it is out of the scope
of our studies.
It is discussed merely because it is really where the intuition of
differentials stems from, and because it is interesting.
6. In terms of real numbers, when we think of a differential it is
essentially 0 (it is arbitrarily close to 0), but in some situations of
using differentials we arrive at finite results. We can arrive at a
finite result in two cases
1. When we look at the ratio of two differentials, the ratio of
infinitessimal changes may no longer be infinitessimal.
2. When we sum an infinite number of infinitessimal quantities, the
sum may no longer be infinitessimal.
Cont...
7. The above statements tell us that although infinitessimals are
arbitrarily close to 0, they are certainly not 0.
First, we cannot divide by zero, and second, no matter how many
zeroes we add together, the result is still zero.
Above we represented the differential df as f(x + dx) f(x), but we
can represent it in another way as well (in this way we can avoid
the difficulties and subtleties of calculating f(x + dx) f(x)).
Cont...
8. Assuming that f'(x) exists, moving an infinitessimal distance along
the curve f(x) is identical to moving an infinitessimal distance
along the tangent line to f at the point x.
This follows from the fact that a differentiable function is
essentially linear in a small enough neighborhood around a given
point, so that by restricting that neighborhood to be small enough,
we can make the tangent line approximation as accurate as we like.
Thus, for a differentiable function,
Cont...
9. Nearly any elementary calculus text will caution us not to simply
cancel the differentials dx and write o ffthe above equation as trivial.
Why shouldn’t we?
Looking at a slightly more complicated expression, the chain rule,
we can find reason for this.
The chain rule states that, for differentiable functions f(u) and u(x),
we have
Cont...
10. One would think that this identity is trivial if we simply cancel the
differentials du. Why can’t we do this? Well let’s suppose that in
general we could cancel the differentials. Now let’s look at the
functions
Cont...
11. These in and of themselves seem to be very esoteric functions, so
let’s take a minute to think about what they really are. For the
function f, whenever the input is a rational number, we get an
output of 1.
Otherwise, for an irrational number, the output is 0. This function is
so badly broken up that we cannot draw it, but we can note that it is
discontinuous everywhere (because between any two rationals is an
irrational, and between any two irrationals is a rational).
Cont...
12. The second function u behaves similarly to f, except that it is 0 for
rationals and 1 for irrationals. It is equally badly behaved.
However, when we look at the composition f(u(x)), something
remarkable happens. No matter the input, u outputs a rational number
(either 0 or 1), so the output of f(u(x)) = 1 for all x. This function is
continuous everywhere, as well as differentiable (f ◦ u)'(x) = 0 for all x.
However, this function does not satisfy the hypothesis of the chain rule,
because neither of the component functions is differentiable
anywhere.
Thus, writing that
Cont...
13. by canceling the differentials du would be a grave mistake, because
neither of the functions on the left-hand side is differentiable, so
the product of two things which do not exist clearly cannot be the
derivative of another function, namely 0!
The above example may feel very artificial, and in some ways it is.
But it does provide us with the reason why there is so much
caution about simply canceling the differentials,
Cont...
14. because it is possible to conceive of circumstances where doing so would
lead us astray. Now that we know we cannot always cancel differentials as
such, the natural question to ask is: when can we cancel differentials? Well,
we know that if f 0(x) exists, then
which tells us that as long as f 0(x) exists in the above situation, we can
effectively cancel the dx’s. Similarly, with the chain rule, if we know that f
'(u) and u'(x) exist (which implies that (f ◦ u)'(x) exists), then
Cont...
15. which says there are multiple ways of representing the differential df.
The above follows
because
by definition. We will close by saying that in most cases where the
derivatives all exist, the differentials usually cancel. Nevertheless, we
caution the reader to first be certain that the differentials really cancel,
by deriving the relationship from
We will finish with a few examples.
df = f'(x)dx
Cont...
16. We will finish with a few examples.
Example 1 Find dy if y(x) = x · cos(x).
Solution Since y is a differentiable function of x, we can use the
above formula, which tells us that dy = (cos(x) x sin(x))dy
Example 2 Find du if u(y) =
Solution Once again we have a differentiable function u(y), so we
find that
Concluion
