Meaning of Second Derivative Notation

Date: 07/08/2004 at 16:44:45
From: Jamie
Subject: second derivative notation

What does the second derivative notation, (d^2*y)/(d*x^2) really mean?  

I understand that the notation in the numerator means the 2nd 
derivative of y, but I fail to understand the notation in the 
denominator.  Isn't it supposed to mean with respect to x?  Why is 
there an x^2 in the notation?

Date: 07/08/2004 at 23:39:22
From: Doctor Peterson
Subject: Re: second derivative notation

Hi, Jamie.

I don't think this is explained nearly as often as it should be!  
There is no x^2 in this notation, and in fact no multiplication (ie,
it is _not_ d*x^2 as you say).  It is

  d^2y
  ----
  dx^2

and the "d" represents the "differential operator", which evidently 
has higher precedence than exponentiation.  That is, "dx" as a whole
is thought of as a quantity (think of it as a small change in x), and
the denominator is "(dx)^2".

But here is where it comes from: the second derivative is just the 
derivative of the derivative, or

  d  dy    d(dy)    d^2y
  --(--) = ------ = ----
  dx dx    (dx)^2   dx^2

You might read it as "the second derivative of y, with respect to x 
TWICE"; that last word is the reason for the "dx^2".  When you have 
functions of more than one variable you can see things like

  d^2z
  -----
  dx dy

(though a modified "d" is used to avoid some confusion); this means 
you are taking one derivative with respect to x and another with 
respect to y:

  d  dz
  --(--)
  dx dy

This notation is based on analogies to fractions, and it can be 
dangerous to imagine that the dx and dy and d alone actually stand 
for numbers; but the notation works very well in making many 
formulas memorable.  See this page for more on differentials:

  Differentials
    http://mathforum.org/library/drmath/view/53678.html 

If you have any further questions, feel free to write back.

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