The meaning of differentiation and its uses:
Differentiatial Calculus allows us to measure 'rates of change' - I'll illustrate this... If you take a normal mathematical function, whether it's a straight line, or a curve, when you plot it you will see a visual representation of the change in a value (y) according to different values of an independant variable (x). This is useful, because you can see that as 'x' increases, 'y' increases, say, twice as fast - that's just an example. A function can really do anything to a number - so something more complex like a polynomial might cause a wavey line, or a u-shaped curve, or something really weird (like tan, or atan).
Anyway, what's all that got to do with differentiation? Well, as I mentioned in the first line, differentiation allows us to measure a 'rate of change' - by this I mean "how fast is y changing with varying values of 'x'?". Measuiring a rate of change is subtly differnet to just measuring change. It allows us to determine more about the nature of a system. Here's a mechanical example:
Say a car's velocity at a given point in time is 60km/h. We also know that the car is accelerating (getting faster) at a fixed rate of acceleration, but we don't know the rate of acceleration. All we know is that 5 seconds later the same car is travelling at 100 m/s. In this circumstance we could plot the velocity of the car against time and the resulting line would illustrate just that - so we can tell at any point in time what the car's velocity would be, but how can we tell what it's rate of acceleration would be? Well, acceleration by definition is the 'rate of change of velocity' - now didn't I just say that differentiation allows us to determine 'rates of change' ? So let's try it:
if accelleration is the 'first derivative' of velocity, then we can express this as:
a = dv/dt (imagine the letter 'd' to be a 'delta' sign, meaning 'change').
that means that acceleration is the 'rate of change of veolcity with respect to time', so let's calculate it:
a = change in velocity / change in time
a = 100 - 60 kmph / 5s
a = 40kmph / 5s
a = 8 kmph / s
That means the car is accelerating by 8 km/h every second.
Now let's tidy up the units:
8km = 8000m
1h = 3600 s
=> a = 2.2 m/s/s
It does seem a bit weird to say 'per second per second', so mathemeticians like to express this as ms^-2 (where ^ means 'raised to the power of').
Okay - I'll take this a little bit further. I mean, this is all a bit silly isn't it? All we have done is measured the gradient of a straight line - that was useless. Well yes, and sometimes the first derivative is referred to as the 'gradient function' for just this reason. But the cool thing to note is that we now know the acceleration at any point along that velocity/time curve. I know, I know - it's not a curve, it's a straight line, but what if it was a curve? What would you do then?
So let's say that the car is on the outside lane of a dead-straight bit of road. The driver floors the gas and speeds off into the distance. You'll probably find that the acceleration of the car isn't actually a straight line - it's more likely to be defined by a curved, which constantly changes gradient as the torque and efficiencies in the engine change through the acceleration phase. It might be important for engineers to know exaclty what the 'gradient function' of this acceleration curve looks like in order to get an instant view of peak accelerations. They would then a) quote these values on their sales papers and b) check that the structure of the car can withstand such acceleration and that it doesn't get pulled to pieces by its own inertia!
This is one example in which we might use differentiation on a 'function' to get more detailed information on the behaviour of the associated system.