Re: Definition of Derivative and its relationship with the concept of differential

Hi there!

We know that the definition of the derivative of a
function nowadays is
made by using the concept of the limit though
traditionally it wasn't
so. Thus, considering a function y=f(x), the limit
when delta x tends
to zero of the incremental quotient delta y over
delta x (delta y =
f(x+delta x) - f(x)) is called y' or f'(x), i.e. the
derivative of
f(x). (1)

On the other hand, the definition of the differential
of a function
y=f(x), is dy=f'(x)·dx. (2)

Now, according to (2) we can algebraically isolate
y'=f'(x)=dy/dx. So,
the derivative can be expressed as a quotient of
differentials.

The questions are:

i) What's the relationship between (1) and (2)? Is
the derivative the
limit of an increment quotient or a quotient of
differentials?
The way you have defined them above, the derivative cannot be DEFINED as a quotient of differentials because you have to use the derivative to define the differential! AFTER you have defined them, then yes, it is true that y'= dy/dx (right side being a quotient of differentials).

If they
both are the same, the limit of the quotient would be
the quotients of
limits (being dy= limit when delta x tends to zero of
delta y, and
dx=limit when delta x tends to zero of delta x),
which is not true
No. dx and dy, the differentials are NOT defined by limits. They are defined by dy= y'(x)dx just as you said.

since this property of limits is only correct if the
denominator is
different from zero.

A differential is never 0.
ii) Is it reasonable to consider a differential of a
differential, e.g.
d(dx)=d^2(x)? What's it equal to? Can it be dismissed
compared to dx?
What's the value of d(dy)=d^2(y)? Is it reasonable to
talk abou the
n-th differential as we talk about the n-th
derivative?

Thanks.

greenishguy.

Yes, it is possible to talk about "the differential of a differential"- that sort of thing is done in differential geometry. However, d(dx)= 0 always. In general d(df)= 0 unless f has singularities. That being the case higher order differentials are necessarily 0 also.

By the way, it is possible, using some very deep concepts from logic, to show that it is possible to extend the real number system to include both "infinitely large" and "infinitely small" numbers (that's called "non-standard analysis"). In that case we CAN define "dx" and "dy" separately and THEN define the differential as the quotient dy/dx. However, dx and dy are not ordinary numbers, they are "infinitesmals". The rules of arithmetic are quite different for infinitesmals and infinite numbers than they are for ordinary numbers.
.

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