Higher Concepts of Mathematics
Using equation 5-1 we find the average velocity from S1 to S2 is
. If we connect the
points S1 and S2 by a straight line we see it does not accurately reflect the slope of the curved
line through all the points between S1 and S2. Similarly, if we look at the average velocity
between time t2 and t3 (a smaller period of time), we see the straight line connecting S2 and S3
more closely follows the curved line. Assuming the time between t3 and t4 is less than between
t2 and t3, the straight line connecting S3 and S4 very closely approximates the curved line between
S3 and S4.
As we further decrease the time interval between successive points, the expression
closely approximates the slope of the displacement curve. As
D t 0,
The expression for the derivative (in this case the slope of the
displacement curve) can be written
. In words, this expression would be
D t o
"the derivative of S with respect to time (t) is the limit of
as Dt approaches 0."
The symbols ds and dt are not products of d and s, or of d and t, as in algebra. Each represents
a single quantity.
They are pronounced "dee-ess" and "dee-tee," respectively.
expressions and the quantities they represent are called differentials. Thus, ds is the differential
of s and dt is the differential of t. These expressions represent incremental changes, where ds
represents an incremental change in distance s, and dt represents an incremental change in time
The combined expression ds/dt is called a derivative; it is the derivative of s with respect to
t. It is read as "dee-ess dee-tee." dz/dx is the derivative of z with respect to x; it is read as
"dee-zee dee-ex." In simplest terms, a derivative expresses the rate of change of one quantity
with respect to another. Thus, ds/dt is the rate of change of distance with respect to time.
Referring to figure 3, the derivative ds/dt is the instantaneous velocity at any chosen point
along the curve. This value of instantaneous velocity is numerically equal to the slope of the
curve at that chosen point.
While the equation for instantaneous velocity, V = ds/dt, may seem like a complicated
expression, it is a familiar relationship. Instantaneous velocity is precisely the value given by
the speedometer of a moving car. Thus, the speedometer gives the value of the rate of change
of distance with respect to time; it gives the derivative of s with respect to t; i.e. it gives the
value of ds/dt.