Example:

An acquisition officer observed the cost of similar defense contracts for the past 10 years. He is interested in the rate of change in the costs for contracts that last 2 years. The following table depicts the values he obtained.

Duration in months	Cost in $1,000
10	33
20	79
30	165
40	236
50	362
60	491

Using these values, he plotted the points on a graph of COST versus DURATION as represented by the red dots in the illustration below.

COST DURATION

Please refer to the graphing functions station for methods for deriving a function from a graph.

From the graph, he was able to find a function (illustrated by the blue line, above) which represents the graph of cost c versus time t. The officer found that the function which best fits the points on the graph is

c = 0.1 t² + 2t
With this function, he was able to compute the rate of change in cost with respect to duration by using the concept of derivatives. The acquisition officer took the derivative c' of the function c
c' = 0.2 t + 2
With the derivative c', he can now determine the rate of change of cost of a contract as it increases beyond two years or 24 months. The derivative c' at time t = 24 months given in $1000's per month is then
c' = 0.2 × 24 + 2 = 2.48

Definition:

What is this derivative and how does it work? Let us see what the definition of derivative is. A derivative f'(x) of a function f depicts how the function f is changing at point x. It is necessary for the function f to be continuous at point x in order for there to be a derivative at that point. A function which has a derivative is said to be differentiable.

The derivative is computed by using the concept of x. x is an arbitrary change or increment in the value of x. You can see that if x in the example above is 10 months, you can readily compute the rate of change in the function, but the officer wanted to know the rate of change at a particular point x, not a span of 10 months. The derivative is the limit approached by the rate of change in the function when x becomes arbitrarily small.

To understand this idea of x consider the group of 10,000 marching men depicted below:

If we add one more man to the group you will probably not be able to notice the difference, because the addition of one man is very small when compared to 10,000 men. If we can measure the rate of change in a function is smaller and smaller increments, we will approach the rate of change at a point. In general, the derivative of function f(x), also called dy/dx can be defined as

This means that as the x gets very small, the difference between the value of the function at x and the value of the function at x + x divided by x is the derivative.

Give x an arbitrary change, x, and compute the new value of the function. Subtract the original value of the function, y = f(x), from the new value, y + y = f(x + x), to obtain the change in the function. Divide the change in the function, y, by the change in x, x, to obtain the average rate of change. Determine the limit, dy/dx, of the average rate of change of the function as x approaches zero