Displacement Versus Time:

Using the graph of displacement, S, versus time, t, in below figure, we will try to elaborates the concept of the derivative.

Figure: Displacement Versus Time

Using equation 5-1 we find out the average velocity from S₁ to S₂ is S₂ - S₁/t₂ -t_1.   If we connect the points S₁and S₂ through a straight line we see it does not accurately reflect the slope of the curved line by all the points among S₁ and S₂. Similarly, if we look at the average velocity among time t₂ and t₃ (a smaller period of time), we see the straight line connecting S₂ and S₃ more closely follows the curved line.  Supposing the time among t₃ and t₄ is less than among t₂ and t₃, the straight line connecting S₃ and S₄ extremely closely approximates the curved line between S₃ and S₄.

As we additionally decrease the time interval among successive points, the expression ΔS/Δt more closely approximates the slope of the displacement curve. As Δt  →0 ΔS/Δt approaches  the instantaneous  velocity. The  expression  for  the  derivative  (in  this  case  the  slope  of  the displacement curve) can be written as dS/dt  = ΔS/Δt In words, this expression would be "the derivative of S along with respect to time (t) is the limit of ΔS/Δt as Δt approaches 0."

The symbols ds & dt are not products of d and s, or of d and t, as in algebra.  Each represents one quantity. They are pronounced "dee-ess" and "dee-tee," respectively. These expressions and the quantities they represent are known as differentials.  Therefore, ds is the differential of s and dt is the differential of t.  These expressions represent incremental changes, whereas ds represent an incremental change in distance s, and dt represents an incremental change in time t.