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### 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

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/
```
Associated Topics:
High School Calculators, Computers

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