# Small question about differential operators

• Apr 8th 2013, 01:32 PM
Geo877
What is the result of $\frac{d}{dx}\left(x\frac{d}{dx}\right)$?

Do you use the product rule or is it simply $x\frac{d^2}{dx^2}$?

Thanks
• Apr 8th 2013, 01:43 PM
Plato
Re: Small question about differential operators
Quote:

Originally Posted by Geo877
What is the result of $\frac{d}{dx}\left(x\frac{d}{dx}\right)$?

Do you use the product rule or is it simply $x\frac{d^2}{dx^2}$?

Thanks

I an from a school of mathematics that does not like that notation.

Having said that, if you must use it then use the product rule.
• Apr 9th 2013, 07:08 AM
Geo877
Re: Small question about differential operators
Thanks Plato, I can't say I'm a big fan of it, but it's rather ubiquitous in physics. So the correct result is:

$x\frac{d^2}{dx^2}+\frac{d}{dx}$
• Apr 9th 2013, 07:56 AM
topsquark
Re: Small question about differential operators
If you are looking at this as a Physicist the typical way to introduce this kind of concept is to use your operator on a function. For example:
$\frac{d}{dx} \left ( x \frac{d}{dx} \right ) f = \frac{d}{dx} \left ( x \frac{df}{dx} \right )$

Use the chain rule:
$\frac{d}{dx} \left ( x \frac{d}{dx} \right ) f = \frac{df}{dx} + x \frac{d^2f}{dx^2}$

And now "factor" the f out:
$\frac{d}{dx} \left ( x \frac{d}{dx} \right ) f = \left ( \frac{d}{dx} + x \frac{d^2}{dx^2} \right ) f$

Thus
$\frac{d}{dx} \left ( x \frac{d}{dx} \right ) = \frac{d}{dx} + x \frac{d^2}{dx^2}$

This is often used for QM and perhaps Mathematical Physics classes until the students learn to "grok" doing this without the function. I admit that the method I did above is little different from just applying the chain rule but Physicists often use it to keep their heads on straight. (I learned it myself, not from one of my Physics classes, but in a text I inherited from a former Professor on Laplace transforms.)

-Dan