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 -t1. 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.