When would you want to model a derivative? I just read this, and am intrigued.
http://formulize.nutonian.com/documentation/eureqa/tutorials/modeling-derivatives/
What kind of model would you have a scatter plot of data points, and want to compute a "rate of change" derivative for ?  I'd like to try this, but need helping in envisioning a "real world" example that I can easily relate to.    Something a little less academic that finding the speed of a car, etc. 
 A: Where wouldn't you need it? Almost all natural laws involve derivatives (it's almost always "rate of change equals something expressed with physical quantities"). I won't start enumerating too many things because we could just list everything, from rates of price changes in retail, dynamics of stock markets, measurements of population growth, disease outbreaks, and of course everything you ever heard of in physics and engineering.
However, because numerical derivative immensely increases noise, it's rarely done this way unless you need to. If you have a model, it's easier to fit to the original data points (not the derivative). Numerical differentiation is used when you have real-time data and you are not waiting for the entire measurement to complete, and when there is no global law you are fitting to (things change dynamically). Most of the examples above are of this type (there's no model function that predicts the curve).
Another a bit different situation is also, that there is an unknown offset in data that you really don't want to deal with and isn't really important for your measurement. If your sensor is giving out something like $g(t)=af(t)+b$ and $b$ is the bogus signal when it should be zero, and $f(t)$ is what you actually wanted to measure, then differentiation gets rid of the offset. For instance, dividing two different signals that measure the same process (so $f(t)$ should be the same), then $g_1(t)/g_2(t)$ won't give you anything userful, but the derivatives $g'_1(t)/g'_2(t)=a_1/a_2$ which tells you the ratio of multipliers.
Real-time differentiation is common in electronics: when you move past a sensor that automatically turns on the light on your porch, it is observing the rate of change in the reflected light intensity, because it only cares about motion, not static obstacles (such as a potted plant that wasn't there yesterday). You can actually differentiate with an analog electronic circuit, no digital chips needed - simply a capacitor and some resistors.
A: Set a hot cup of coffee on the counter, and put your digital thermometer in the cup. Every minute record the temperature of the coffee. You might want to know how fast the temperature of the coffee was changing in the 10th minute of this experiment. You could then compare that to how fast the temperature of the coffee was changing in the 30th minute of the experiment. (Keep in mind, I am talking about the rate at which the temperature of the coffee is changing, in say $^\circ$ F/min, not the temperature (in $^\circ$ F) of the coffee).
Now picture taking those readings every 30 seconds instead of every minute. Then every 10 seconds. Then every second. Then every tenth of a second, and so on. The temperature of the coffee is a function of continuous time, and your readings are just taking discrete readings from that function, but they can be used to approximate the true instantaneous rate of change at a given time $t$ by letting $\Delta t$ (the time gap beyween your readings) get smaller and smaller and smaller, i.e. the limit as $\Delta t\to 0$. This is why the idea of the limit is so fundamental to the notion of a derivative.
Once you understand the difference between the rate of change of a quantity in contrast to the quantity itself, you can't help but see rates of change all around you! Some other examples:


*

*$\displaystyle{dT\over dt}$ rate of change of the temperature $T$ with respect to time $t$

*$\displaystyle{dN\over dp}$ rate of change of the number of tickets $N$ sold at a concert as we vary the price $p$ in dollars

*$\displaystyle{dV\over dr}$ rate of change of the volume $V$ of water leaking from a bucket was we vary the radius $r$ of a circular hole poked in the bottom

*$\displaystyle{dE\over dv}$ rate of change of the fuel economy $E$ (in say mpg) in your vehicle with respect to the velocity $v$ (in mph) that you drive

*$\displaystyle{dg\over dh}$ rate of change of your grade $g$ in calculus class as you vary the number of hours $h$ spent studying

