(_width_) EQUALS (_floor area_) DIVIDED BY (_length_) The first equation shows that the floor area depends on the length of the room and also on the width of the room. So we say floor area is a _function_ of length and width. This particular function happens to be _product_, the result of multiplication. In other words, floor area is equal to the product of length and width. Now there is another kind of function called a _differential function_ or _derivative_. A _differential function_ or _derivative_ is an _instantaneous rate of change_. An instantaneous rate of change is the result of two steps: (1) finding a rate of change over an _interval_ and then (2) letting the interval become smaller and smaller indefinitely. For example, suppose that we have the problem: How are speed, distance, and time related to each other? One of the answers is: (_speed_) EQUALS THE INSTANTANEOUS RATE OF CHANGE OF (_distance_) WITH RESPECT TO (_time_) Or we can say, and it is just the same thing in other words: (_speed_) EQUALS THE DERIVATIVE OF (_distance_) WITH RESPECT TO (_time_) Now we can tell what a differential equation is. It is simply an equation in which a derivative occurs, such as the last example. Perhaps the commonest kind of equation in physical problems is the differential equation. SOLVING PHYSICAL PROBLEMS Now we were able to change the equation about floor area into other forms, if we wanted to find length or width instead of floor area. When we did this, we ran into the _inverse_ or opposite of multiplication: division. In the same way, we can change the equation about speed into other forms, if we want to find distance or time instead of speed. If we do this, we run into a new idea, the inverse or opposite of the derivative, called _integral_. The two new equations are: (_distance_) EQUALS THE INTEGRAL OF (_speed_) WITH RESPECT TO (_time_) (_time_) EQUALS THE INTEGRAL OF [ONE DIVIDED BY (_speed_)] WITH RESPECT TO (_distance_) These equations may also be called differential equations. An integral is the result of a process called _integrating_. To integrate speed and get distance is the result of three steps: (1) breaking up an interval of time into a large number of small bits, (2) adding up all the small distances that we get by taking each bit of time and multiplying by the speed which applied in that bit of time, and (3) letting the bits of time get smaller and smaller, and letting the number of them get larger and larger, indefinitely. In other words, (_total distance_) EQUALS THE SUM OF ALL THE SMALL (_distances_), EACH EQUAL TO: A BIT OF (_time_) MULTIPLIED BY THE (_speed_) APPLYING TO THAT BIT This is another way of saying as before, (_distance_) EQUALS THE INTEGRAL OF (_speed_) WITH RESPECT TO (_time_) To solve a differential equation, we almost always need to integrate one or more quantities.