# Derivative Function

We defined the derivative at a point. If we take the collection of all the derivatives of all the points of a function, we wind up with a new function. This derivative function tells us the value of the derivative for any point on the original function. We define the derivative function as: This definition yields a function. When we evaluate the derivative function for a given x value, we get a number which is the derivative at a point (i.e., the rate of change of f, or the slope of the graph of f ).

If y = f (x), an alternate notation for the derivative function is which is reminiscent of the difference quotient "change in y over change in x." A third notation uses as a derivative operator. For example f (x) This is different from operators like + and -, which take numbers and give a number as the result. This operator however takes a function and gives a new function (the derivative) as a result. When evaluating a derivative function at a specific point, such as x = 2, you use either f ' (2) or . This device cannot display Java animations. The above is a substitute static image
 See About the calculus applets for operating instructions.

## 1. A Parabola

The applet initially shows a parabola on the left and the derivative function of the parabola on the right. At the bottom of the applet is a slider which controls the x coordinate, which is displayed in an input box next to the slider. On the left-hand graph is a red line which represents the tangent line at the x coordinate. Move the slider and note that the tangent lines moves so that it is always tangent to the parabola at the x coordinate specified by the slider. At the bottom left-hand corner of the function graph is a box that gives the value of the function f (x).

Now look at the right graph, which shows the derivative function, f ' (x). First, look at the red tangent line; what is its slope? Its slope must be the derivative at the current x coordinate, so that must also be the value of the derivative function for that x coordinate. This slope is shown in a box at the lower left-hand corner of the derivative graph. The point on the graph of the derivative function is also noted by a red crosshair.

Click in the "x=" box and replace its contents with 0. Now drag the slider to the right. Notice that, as the slope of the red tangent line increases, the derivative function also increases. Drag the slider to the left past 0. Note that as the slope of the red tangent line becomes more negative, so does the derivative function. The derivative function tells you the rate of change of f for any given x, which is equivalent to telling you the slope of the graph of f for any given x.

When the derivative is positive, the function is increasing. When the derivative is negative, the function is decreasing. Hence the derivative tells you something about the original function. What happens when the derivative is 0? Where does this happen in this example? Why is the derivative 0 at that point?

Notice also that the derivative function looks like a straight line. Do you think this will always be the case, or is this due to some special property of parabolas?

## 2. A sine function

Select the second example from the drop down menu, showing a sine function. What does the derivative function look like? Drag the slider, watch the slope of the red tangent line, and see if you can relate the slope of the tangent line to the value of the derivative function. Is the derivative 0 at any points? What characterizes those points?

## 3. An exponential function

Select the third example, showing an exponential function. What does the derivative function look like? Drag the slider, watch the slope of the red tangent line, and see if you can relate the slope of the tangent line to the value of the derivative function. Note that for the exponential function, its derivative function is never negative (i.e., the right-hand graph never drops below the x-axis). Why? What is it about the exponential function's graph that means the derivative is never negative?

## 4. A hyperbola

Select the fourth example, showing a hyperbola. What does the derivative function look like? Drag the slider, watch the slope of the red tangent line, and see if you can relate the slope of the tangent line to the value of the derivative function. Note that for this hyperbola, its derivative function is never positive (i.e., the right-hand graph never rises above the x-axis). Why? What is it about the hyperbola's graph that means the derivative is never positive?

What happens at x = 0 for the hyperbola? Why is the derivative undefined? What is the slope of the tangent line (is there a tangent line)?

## Explore

You can also type your own function definition into the "f(x)=" box to see what the derivative of other functions look like.