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Let's imagine a body constituted by the only mass unit i(0) placed in the coordinates´ origin E1(0), E2(0), E3(0),.... En(0), that makes the same decisions according to an only stimulus provided by a mass unit i(1) with coordinates E1(1),E2(1),E3(1),....En(1),
En (0)= En(1) for values n=2 to n=n
And also, being all the decisions made in the face of the same proposal equal, if we assume that decision which remains the same is "a" of which quantity in the X axis.
Therefore this situation would correspond to the most basic model that we could imagine and it can be represented inside the same coordinate axis.
Suppose that in t time the i(0) unit has had a number of decisions E1,x according to i(1) Point A in the axis E1,x
Later in t' time, it has had x' decisions. Point B in the axis E1,x
PA= E1, x
PB= E1, x'
Is the increase of the decisions that the i(0) unit had in the time increase,
The average speed of the i(0) unit according to the i(1) unit is defined by:
Consequently the average speed during certain time gap is the same as the average decisions made by each time unit. In order to settle the instantaneous speed in a certain place, for instance A, we have to make the time gap Dt as short as we can, so that essentially there won't be any changes in the state of the choices during that small gap. In the mathematics language that means to, calculate the value of the previous fraction limit when the denominator Dt tends to zero.
That is the definition of derivative of s according to time t.