The derivative of a function is one of the two central objects of study in calculus. The integral is the other (we will see this later). All objects in mathematics have definitions. In this section we give the definition of the derivative of a function. This, of course, gives us the understanding of the derivative. Without this understanding we can not expect to use the derivative effectively.
Up to this point, we have been interested in the function’s values or the limiting behavior of the function’s values. The derivative of a function is another function whose values measure the rate of change of the original function’s values. For a linear function this will be a constant function whose value is the slope of the straight line. To get some understanding of the derivative consider the odometer and speedometer in a car. The odometer can be considered to measure the distance that the car travels as a function of time - i.e. the odometer has one value for each time. The speedometer measures how fast the car is traveling as a function of time. The speedometer reading can then be interpreted as how fast the distance is changing in time. So where as the odometer measures in miles, the speedometer measures in miles per hour which is the rate of change of distance with respect to time.
Average rate of change.
If you computed the average speed ( average rate of change ) of a car trip you would proceed as follows. First you would record the odometer reading and the time at the beginning of the trip. Then at the end of the trip you would record the final odometer reading and the time. You would subtract the beginning odometer reading from the final odometer reading to get the total distance traveled on the trip. Then you would subtract the time at the beginning of the trip from the time at the end to get the total time of the trip. You would then divide the total distance traveled by the total time of the trip. This would be the average speed during the trip. More precisely, it would be the average rate of change of the odometer reading ( a function of time ) with respect to time.
Suppose, as an example, that the odometer initially read 32,763 and the final reading was 32,773. The trip started at 3:00 PM and ended at 3:30 PM. So, this was a 10 mile trip that took one-half hour.
The average speed is 20 mph. During the trip you probably did not travel 20 mph all of the time. Sometimes you went faster and sometimes slower. The speed that you were traveling at any particular time was measured by the speedometer. If we did the same computation at smaller and smaller time intervals, we would expect that the average speed would be close to the speedometer reading. After all, how much can the speed of a car change in say one millisecond? The speedometer reading appears to be a limit of the average speed as the time interval ( final - initial ) gets smaller and smaller. This is the idea behind the derivative. This example gives a physical interpretation of the derivative.
We now give a geometrical interpretation which will serve to motivate the definition of the derivative. This will be the slope of the tangent line to the graph of a function at a point on the graph. We will then see that this is also the rate of change of the function values.
There's not enough information to compute the slope of a line given just one point on the line. We will use two points as dictated by the slope formula, the first point is given, the second is arbitrary. We then take the limit as the second point moves towards the first. This will give us a well-defined procedure for computing the slope as well as defining what the tangent line is. We will refer to the diagram that follows.
Notice that the last formula depends on x and delta x. We are thinking of x as fixed and so the slope is a function of delta x. Now consider the following picture. The idea is that the second point we used to compute the slope was arbitrary. So different people could choose different points and hence get different values for the slope. In order to eliminate this arbitrariness we will get the limiting behavior of the slope as delta x gets smaller and smaller. Hopefully the following picture will help you to see what's going on.
What we are trying to see here is that as delta x goes to zero, the points on the graph, starting with the right most, move towards (x,f(x)). The lines through these points move also but less and less as delta x gets smaller and smaller. There is a limiting line. This line will be called the tangent line and its slope is, by definition the limit of the slope formula given above. The graph of the function and its tangent line are as follows.
The slope of the tangent line is then given by the following formula.
We can compute the right hand side for any function regardless of the geometric interpretation. So this is strictly a mathematical construct. As such we will call it the derivative of f(x) with respect to x. It is itself a function of x. We will denote this function by f ' (x). Here is the definition that you should remember.
The Definition of the Derivative |
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It's very important to realize that the derivative of a function is another function. The values of the derivative function measure the rate of change of the values of the original function. When the original function's values measure distance as a function of time, the derivative's values measure velocity. When we are thinking geometrically, the derivative's values measure the slope of the tangent line (by definition). There are many other interpretations that you will run into in your other courses. We will only be interested in the mathematical definition.
Let's look at an example to see how to compute the derivative of a function from the given function using the definition . After we do this here (and one time on the first test) we will compute the derivatives in another way. This new way will be easier than this way but still not easy. The reason we are doing it the long way here is that all of the formulas that we use later come from the definition of the derivative. This will help us to appreciate that fact.
We will compute the derivative of
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We have the following results. 
Let's draw the graphs of these functions to see what f ' (x) says about f(x).
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We have f(-1) = 1 and f ' (-1) = -2. Notice that the slope at (-1,1) on the graph of f(x) can be estimated using the triangle as 4/(-2) = -2. The value of f ' (-1) = -2 which is the same, as it should be. The value of f '(x) is the slope of the tangent line to the graph of f(x). That's the geometric interpretation. It is always the rate of change of the values of f(x) with respect to x. When x changes f(x) changes and f '(x) measures how fast the values of f(x) are changing. This is, of course, what the slope measures for linear functions. If this does not make sense to you, then it's time to review the section on the qualitative aspects of straight line functions (linear functions).
The derivative is for functions whose graphs are not straight lines what the slope is for straight lines. The "slope" of a nonlinear (not straight line) function changes from point to point so we need a function to say what the slope is at a point. The value of f ' (x) is the rate of change of f(x) at the point (x,f(x)). Geometrically, that is the slope of the tangent line through the point (x,f(x)), not by magic but by definition. It is impossible to understand something if you don't know its definition. Reread all of the above if you don't see where the definition of the derivative applies.
Notation
We use the following two notations for exactly the same thing - the derivative of the function f(x). I typically use the first when I just have the name of the function. The second I use when I have my hands on the formula for the function.
Other people might use some of the following.
They all mean exactly the same thing.