A Tabular View of the Derivative
We have looked at the derivative function from a graphical point of view.
Let's now look at approximating the derivative function using a table of
values.
Try the following:
- The applet shows a table that represents selected values of a function.
The first row shows a selection of x values, while the second row
shows the corresponding values for f (x). The third row
shows estimated values for the derivative. These are computed from the
pair of x values and the pair of y = f (x)
values above each estimate from the difference quotient
. Note
that this is only an estimate for the derivative, as this is calculating
the average rate of change on the interval, not the instantaneous rate of
change. But, since we only have selected values for x and f (x), that's the best we can do just using the data in the table.
For example, the first derivative estimate is computed as 
and the
next value is 
. Can you calculate some of the other values for the
estimated derivative, using the data in the table?
- Select the second example from the drop down menu. This is the same
function, but now the x intervals are farther apart. The
derivative is still estimated using the difference quotient, but in this
example the denominator is 2 instead of 1.
- Select the third example, a linear function. Compute one or two
derivative estimates and check that your calculations match those in the
table. What do you notice about the derivative?
- Select the fourth example, a constant function. Why is the derivative
always 0?
- Select the fifth example, an exponential function. Does there seem to
be some relationship between the derivative and the value of the
function? We will return to this in the future.
- Select the sixth example, a hyperbola. Why are the derivative estimates
negative?
- Select the seventh example, a sine curve. Does it make sense for the values of the derivative to change around like they seem to do? Is there an upper and lower bound on the values of the derivative function?
- You can make your own example by typing a function definition into the
"f(x)=" box. You can also select the starting x value and the step
size between x values.
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.