We learned in an earlier lesson that the average velocity of a free-falling object over an interval of time Dt is given by the distance traveled during that interval divided by the time interval.

The instantaneous velocity at time t1 is found by evaluating the limit as Dt→0.

Note that this is just the derivative, which we can write using the limit equation.

Because the velocity is the rate of change of distance with respect to time, the derivative can be thought of as a measure of the instantaneous rate of change.

Let's look at the rate of change of y = f(x) over the interval x1 to x2. The change in x is Dx = x2 - x1. The change in y is Dy = f(x2) - f(x1).

The variables x2 and Dx can be rewritten in terms of only x1.

The instantaneous rate of change of y with respect to x at x1 is then given by the limit as Dx→0. Therefore, the instantaneous rate of change is the derivative f'(x1).

Let's look at a graph of a function. The derivative of a point on the curve is equal to the slope of the tangent line at that point.

Therefore, the instantaneous rate of change at the point is given by the slope.

Now, use the slider to move the tangent line along the curve. Note how the slope, and therefore, the rate of change changes along the curve in the regions A, B and C.

Over what range of x is the rate of change of y the greatest? Click the "Submit" button after making your determination.

The total revenue generated by the sale of a product is modeled by the given revenue function.

The derivative of this function at a certain production level x1 units provides the instantaneous rate of change of revenue at that level. This is known as the marginal revenue.

What is the marginal revenue at a production level of 300 units? Click the "Submit" button after entering your answer.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education