See also: derivative - I didn't see this until after I had created this node.
In calculus differentiation is the process of writing explicitly how an equation differs or changes its value when the variable (or one of the variables) change.
A simple example is the one of the clock. First imagine a old-time clock made of gears and such. One imagine that as the minute hand is moved, the second hand (the one that counts seconds) will have to move 60 times as fast in terms of the angle traveled. The hour hand, on the other hand (pun intended) will only move 1/60th of the angle that the minute hand moves. While only a simple example, differentiation is just this exact thing applied do a host of different problems.
Along with the other fundamental concepts of calculus, differentiation was codified by Isaac Newton. Differentiation is considerably easier to do than integration because there is a formula to follow to achieve the result and there are now in general a set of rules for how to differentiate most functions.
The fundamental definition of the derivative is: df(x)/(dx) = limit as h goes to zero of (f(x+h) - f(x))/h) All the other 'rules' can be derived from this fundamental concept. The short-hand notation is either f with a dot over it or f'
Constants: Constants don't change, the derivative is zero.
Powers: dxn/(dx) = n*xn-1
Linear Combination: (f(x) + g(x))' = f'(x) + g'(x)
Product rule: (f(x)g(x))' = g(x)f'(x) + f(x)g'(x)
Reciprocal rule: (1/f(x))' = -f'(x)/(f(x)^2)
The Quotient rule: (f(x)/g(x))' = g(x)f'(x) - f(x)g'(x) * 1/g(x)^2
There are quite a few other rules but I think all of them can be derived from these . . .