how do you derive Kirchhoff's Current Law?

Question

yeah i'm supposed to get current (i)2 = [R3/(R2+R3)] * (i)1

Answer 1

It sounds as if you're trying to apply the law rather than derive it. If you want the derivation, see ref. 1. Basically it says that a node (basically a junction of two or more wires) can't create charge or make it disappear, so the currents into and out of the node must sum to zero.
If you're trying to use the law to show how the current is divided between two parallel resistors, that's just algebra. If there are two resistors R1 and R2 carrying currents I2 and I3 respectively, and I1 is the total current, we can say
I1 = I2 + I3 (Kirchoff's law)
I2 = E/R2, I3 = E/R3 (Ohm's law)
(where E is the voltage across the parallel combination of R2 and R3)
Thus
I2 = I1-I3 = I1-E/R3 = I1*(1-(E/I1)/R3) (eq. 1)
E/I1 = E/(I2+I3) = E/(E/R2+E/R3) = 1/(1/R2+1/R3) = R2R3/(R2+R3) (eq. 2)
Substituting (2) into (1),
I2 = I1*(1-R2/(R2+R3)) = I1*R3/(R2+R3))