Sometimes it's easy to forget that the derivative of a function is
a function in its own right. It has a graph. You can find its value
at different inputs. It might have a formula. Just as you can ask
where the function is increasing and decreasing, you can ask where
the * derivative* is increasing and decreasing.

The following applet shows both the graph of a function and the
graph of its derivative. It also gives a formula for the derivative,
and it shows a tangent line on the graph of as well as
the corresponding point on the graph of .
**You can change the position
of the tangent line and point by clicking-and-dragging on either graph.**

Use this applet to look at the derivatives of several functions, including: , , , , , , , and . Answer the following questions:

**(a)** What happens on the graph of when the tangent line to
is horizontal? Why?

**(b)** What happens to the tangent line to the graph of when the
point lies above the x-axis? Why?

**(c)** What is the derivative of a linear function such as
?
Explain why this is true,
based on the fact that the derivative of a function gives the slope of that
function at any given point.

**(d)** If you drag the mouse from left-to-right on the graph, the tangent line
seems to rotate, either clockwise or counterclockwise. Can you relate the direction
of rotation to some property of the graph of , or of the graph of , or both?
If you can find the connection, try to explain why it holds.

This question continues the discussion of taxes that was begun in the previous lab. Here is the same table that you looked at last week: The instructions for Form 1040 (Federal Income Tax) include the following table for computing the amount of tax, based on your taxable income. (This is the table for use by a single taxpayer).

If income is over | But not over | Then tax is | of amount over |

$0 | $26,250 | 15% | $0 |

26,250 | 63,550 | $3,937.50 + 28% | 26,250 |

63,550 | 132,600 | $14,381.50 + 31% | 63,550 |

132,600 | 288,350 | $35,787.00 + 36% | 132,600 |

288,350 | $91,857.00 + 39.6% | 288,350 |

**(a)** This table defines a function, , where is the
amount of tax paid on a taxable income of dollars. What is the
derivative, , of this function? Write your answer as a split
function. (Note that on each of the intervals in its definition,
is a linear function.)

For this exercise, you will work with an applet that you've seen before. It shows the graphs of two functions, and , together with the graph of their composition, . This time, however, the applet also shows some tangent lines. To open the applet in a separate, resizable window, click this button:

To use the applet, you can drag the red square on the graph of . This sets the input value, . The applet shows the tangent lines to the three curves at the points , , and . Try moving the red square to see how everything changes. Try some other functions. Try turning on ``Use Mouse'' for one or both functions, if you like. (In this mode, you can define the functions by dragging points on the graph up and down.) When you feel that you know what is going on, do the following exercises.

**(a)** Set up the applet so that both and are linear
functions, such as
and
. Observe the
slopes of the tangent lines. Do this for three or more pairs of linear
functions. How does the slope of the composition compare
to the slopes of the functions and ?

**(b)** Suppose that
and
, where ,
, , and are constants. Calculate a formula for .
Based on that formula, what is the slope of the graph of ?
Does the answer agree with your observation from part (a)?

**(c)** Now, turn from lines to more general functions. For a line,
the derivative is a constant and the value of the derivative is the
same as the slope of the line. For most functions, however, the derivative
varies from one point to another. However, you can still compare the
slopes of the tangent lines for any given input value.

Look at several pairs of functions. For each pair of functions, look at several different x-values by dragging the red square to several different positions. In each case, record the slopes of the tangent lines to , , and . Check whether you get the same sort of pattern that you saw in part (a). (For example, you can try this for and , for , and for and .)

**(d)** Suppose that
. Based on your observations,
what would be the formula for , in terms of the derivatives
of and . (Remember that to get a slope out of a derivative,
you have to plug in an input value. The input value in
is . Be careful to say which input values are going into the
derivatives of and in your formulas.)