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, as in f (x). This is different from operators like + and -, which take numbers
and give a number as the result; 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 .
Try the following:
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?
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?
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?
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)?
You can also type your own function definition into the "f(x)=" box to
see what the derivative of other functions look like.