If it helps you to see this as a very difficult challenge, then go
ahead. I know what the "H" in RAH stands for, but if designing your own
circuit from discrete transistors that you've carved out of wood is
beyond your interest, then why not look around for a plug 'n' play
option? Have you *really* already done that and come up empty handed?
I've got a shelf full of catalogs of temperature control hardware. I
haven't actually looked for one with a rate alarm option, but since rate
is monitored and used by the controller, I don't see why someone
wouldn't offer such an alarm as an option.

Ok, Smitty.

Nope, not at all interested in carving transistors out of wood.

Since you have all the catalogs, my suggestion is to "go for it".

When you are all done, we can compare "budgets" and judge accordingly.
Budgets?

First, obviously, would be cost.
I'm pretty sure, should he decide to bite, Jim's solution would come in
for under 50 bucks.

Power budget?
We can assume we have a 12 volt DC electrical system. How many amps
will you need? Jim probably wouldn't need over 100 mA or so, if that.

Weight budget? (Oh Sh*t, not that again!)
What say, Jim? 3 or 4 ounces?

And then come the electrical issues!

It's a thermocouple. Output is in the millivolt range.
AND, to monitor RATE of change over a very short time (seconds) we'd
have to get down to the microvolt level!

To be able to use that, it will have to be scaled up - gain of roughly
one million or more? No freakin' way *I* would try that with desecrate
(or even wooden) transistors.

Again, it's a thermocouple!
How do you propose to "interface" to it? Single side ain't gonna get it.
You would wind up amplifying the noise as much or more than the signal!
Differential low noise high gain op amp seem to be the best solution here.

Nope, this simple circuit is way beyond anything this ol' logic mechanic
would be able to do. And, not ashamed to admit it, either.

So, let's see whatcha got there Smitty?

Richard