Combined expression:

The combined expression ds/dt is known as 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 easiest terms, a derivative expresses the rate of change of one quantity with respect to another. Therefore, 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 selected point along the curve.  This value of instantaneous velocity is numerically equal to the slope of the curve at that chosen point.

Although the equation for instantaneous velocity, V =  ds/dt,  may  look  like  a  complicated expression, it is a familiar relationship. Instantaneous velocity is precisely the value given through the speedometer of a moving car. Therefore, the speedometer provides 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.

The ideas of differentials and derivatives are fundamental to the application of mathematics to dynamic systems. They are used not only to express relationships among distance traveled, elapsed time and velocity, but also to express relationships among several different physical quantities. One of the most significant parts of understanding these ideas is having a physical interpretation of their meaning. For example, while a relationship is written using a differential or a derivative, the physical meaning in terms of incremental changes or rates of modification should be readily understood.

While expressions are written using deltas, they can be understood in terms of changes. Thus, the expression ΔT, where T is the symbol for temperature, represents a change in temperature. As previously described, a lower case delta, d, is used to represent very little changes.  Thus, dT represents a extremely small change in temperature. The fractional change in a physical quantity is the change divided by the value of the quantity. Thus, dT is an incremental change in temperature, and dT /T is a fractional change in temperature.  Although expressions are written as derivatives, they can be understood in terms of rates of change. Therefore, dT /dt is the rate of change of temperature along with respect to time.