## Quotient rule with determinants?

I’ve always found the quotient rule for differentiation hard to remember:

$$\dod{}{x}\frac{f}{g} = \frac{f’g – g’f}{g^2}$$

I was looking on the Wikipedia page for the finite difference operator and noticed that they use the determinant in describing the quotient rule for the difference operator. The determinant can also be used to describe the quotient rule for the derivative:

$$\dod{}{x}\frac{f}{g} = \frac{1}{g^2}\begin{vmatrix}f’ & g’ \\f & g\end{vmatrix}$$

I wonder why this is? Is it just a coincidence?

**Update:** It occurred to me that it can also be written this way:

$$\dod{}{x}\frac{f}{g} = \det\left(\frac{1}{g}\begin{bmatrix}f’ & g’ \\f & g\end{bmatrix}\right)$$

Since the determinant represents the area of a parallelogram, the derivative at $x_0$ is the area of parallelogram formed by $ \langle f'(x_0),\ g'(x_0)\rangle $ and $ \langle f(x_0),\ g(x_0)\rangle $ where $ g(x_0) $ is scaled to 1. Still, this interpretation is kind of forced. What does it mean?

## Forwarding all email to a domain with `sendmail`

I just bought a new domain name (bitsofpancake.com) — I’m actually quite proud of that name, to be honest! I decided that I would forward all the email addressed to it (whatever@bitsofpancake.com) to my personal email. I had previously done that with a convoluted combination of Google Apps and its catch-all email feature, but this should be much less roundabout. Or so I thought: figuring out how to set it up was a pain.

First, I needed to allow `sendmail` to function as an external mail server.

I’m on Ubuntu Natty, and the `sendmail` config files are located in `/etc/mail/`. It turns out that the `sendmail.conf` is so complicated that there’s a separate file, `sendmail.mc`, that is designed to be edited and then “compiled” into the real config file.

I had to change this line in `sendmail.mc`:

DAEMON_OPTIONS(`Family=inet, Name=MTA-v4, Port=smtp, Addr=127.0.0.1')dnl

into:

DAEMON_OPTIONS(`Family=inet, Name=MTA-v4, Port=smtp')dnl

This change tells `sendmail` to accept external email (source).

Next, I had to enable the virtual user table. I added this line to the same file (`sendmail.mc`) (source):

FEATURE(`virtusertable',`hash -o /etc/mail/virtusertable.db')dnl

I made a file in `/etc/mail/` called `virtusertable` with this one line:

@bitsofpancake.com forwardtothisemail@gmail.com

This catches all emails to @bitsofpancake.com and forwards it to my Gmail account. You can add more lines to this file to specify different forwarding rules (source).

I compiled the `virtusertable` using:

`/etc/mail$ sudo makemap hash virtusertable < virtusertable`

Next, I added bitsofpancake.com to `/etc/mail/local-host-names` — this file tells `sendmail` which domains to consider "local." (Otherwise, it would keep on trying to send to itself in an infinite loop!)

Then, I reloaded and "compiled" `sendmail`'s config with:

`/etc/mail$ sudo sendmailconfig`

Finally, I added port 25 (the port for SMTP) to my `iptables` firewall and updated my MX DNS records to point to the server — it works!