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Differential Notation

Date: 04/16/2009 at 05:50:59
From: Jordan
Subject: Differential Notation

Why is the second derivative written as d^2y/dx^2?



Date: 04/16/2009 at 13:54:14
From: Doctor Vogler
Subject: Re: Differential Notation

Hi Jordan,

Thanks for writing to Dr. Math.  It seems mysterious how the numerator
repeats the d but the denominator repeats the dx, but it makes sense
if you think about it the right way.

You are familiar with the notation

  dy
  --
  dx

to mean "the derivative of y with respect to x".  Sometimes, when you
want to use a formula instead of a variable, you can put that in the
place of y, as in

  d(cos x)
  --------
     dx

or, more conveniently (especially for complicated formulas),

  d
  --(cos x).
  dx

So then if that is the derivative of cos x, then what is its
derivative?  Naturally, it would be

  d   d
  --( --(cos x) ).
  dx  dx

Does that make sense?  And when you write it this way, it is natural
that one would abbreviate this by using the "squaring" notation on the
d in the numerator and the dx in the denominator, as in

  d^2
  ----(cos x).
  dx^2

Of course, you're not really squaring, because it's not
multiplication; but it's a convenient notation, especially when the 2
becomes 12 or an unspecified n.  Similarly, the derivative of dy/dx
would be

  d  dy
  --(--)
  dx dx

which is why you get d^2y/dx^2.

I should like to point out that we are *NOT* squaring the x in the
denominator, like d/d(x^2), but are squaring the dx.  It might have
been better if we wrote (dx)^2 instead of dx^2, but that is not the
notation in common usage by mathematicians, who instead treat the "dx"
as a single piece.

Does this explanation make sense?  I hope I have cleared things up for
you.  If you have any questions about this or need more help, please
write back, and I will try to offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 04/16/2009 at 16:32:20
From: Jordan
Subject: Thank you (Differential Notation)

Comprehensive and laconic answer - thanks ever so much!
Associated Topics:
High School Calculus

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