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The derivative and simple integrals.

1.06 Velocity: the First Derivative

Consider the position of a BODY that moves in 0XYZ - space. At a time (no time specifically but a time, t = t*) an implicit mathematical statement of that position is P(t*) (below left).

This the implicit position of a BODY at a time, t*, in 
space described by the coordinates, 0XYZ.
This the new (implicitly stated) position of the same 
BODY a short time later (at the time, t* + Δt)
in the space described by coordinates, 0XYZ.

Our interest is motion of the BODY until a time later, the inspecific time, t = t* + Δt where the notation Δt is called "an increment in time." By symbol, the position of the BODY at the later time is written (above right).