Re: derivative notation

From: Randy Poe (poespam-trap_at_yahoo.com)
Date: 01/24/05 

Date: 24 Jan 2005 13:20:34 -0800

Vec wrote:
> Hello
> I am having a problem understanding what is going on with this
notation.
> In leaning about derivatives.
> f'(x) = y' = df/dx = d/dx (f(x)) = d/dx (y)
> Why does it break up df/dx to d/dx (f(x)) since the nominator is a
one
> whole notation "df" and that is not d multiply by f because there
is not
> "d" and "f" variables by them selves

Well, df and dx aren't variables either. However, perhaps
you were introduced to them as "differentials", as
the limits of finite intervals, and thus to think
of df/dx as a fraction with a numerator and a
denominator.

Instead, you are now being asked to think of d/dx as a
single thing, an operation (derivative with respect
to x) which transforms a function into another function.
It's not a fraction, not a ratio, just a transformation
rule. It could just as easily be written with some
other notation like Dx[f] instead of the slash that makes
you think of a fraction.

The prime (') notation is more like this operator idea
than like the concept of dividing one thing by another.

When I took calculus 1, I think I was introduced to
differentiation before thinking of dy and dx as
separable things, and I remember reacting violently
to the idea that suddenly these things could be
treated as separate quantities. My teacher said it
was the most vocal I'd been all year.

              - Randy

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