Graphical Understanding of Derivatives:

A function expresses a relationship among two or more variables.  For instance, the distance traveled through a moving body is a function of the body's velocity and the elapsed time.  Whenever a functional relationship is presented in graphical form, an significant understanding of the meaning of derivatives can be developed.

Figure is a graph of the distance traveled through an object as a function of the elapsed time.  The functional relationship display is given by the subsequent equation:

s = 40t

The  instantaneous  velocity  v,  that  is  the  velocity  at  a  given  instant  of  time,  equals  the derivative of the distance traveled with respect to time, ds/dt.  It is the rate of change of s with respect to t.

Figure: Graph of Distance vs. Time

The value of the derivative ds/dt for the case plotted in Figure can be understood by considering little changes in the two variables s and t.

Δs/Δt = (s+Δs) - s/(t +Δt) -t

The values of (s + Δs) and s in words of (t +t) and t, using Equation 4 can now be substituted within this expression.  At time t, s = 40t; at time t + Δt, s +Δs = 40(t + Δt).

Δs/Δt = 40(t + Δt) - 40t/(t + Δt) - t

Δs/Δt = 40t +40(Δt) - 40t/t + Δt - t

Δs/Δt = 40(Δt)/Δt

Δs/Δt = 40