# 9.2 Graphing the Derivative

The spreadsheet construction above gives the user the ability to find the derivative of a function at one specific argument. We want to do the same thing at many different arguments, which can be turned into a chart or graph of the derivative function.

This can be accomplished by picking a single value of $$d$$, putting the structure described in Section 9.1 all on one row, and copying that row down. Now each row will correspond to an argument $$x$$ that increases by $$q$$ from that of the previous row. If we compute both D and E we can compare them. The difference between what is in D and E is a measure of how bad the D estimate is. If it is too big for what we want we can reduce our $$d$$ until we like the result.

OK, how?

Here is an outline of how to do this. It consists of a list of columns and what to put in them.

Suppose you want to graph the value and derivative of a function, say $$\sin(x)$$ from $$x = 0$$ to $$x = 5$$.

You will probably want to put the information: Graphing $$f(x)$$ and $$f'(x)$$ in A1 and $$\sin(x)$$ in A2. In A3 put: starting argument; and in B3 enter $$0$$, in A4 enter ending argument; and in B4 enter $$5$$. In A5 enter number of arguments; and in B5 put your favorite number, say $$100$$. In C5 put =(B4-B3)/B5; In A6 enter $$d$$ and in B6 enter $$0.01$$. Also make the following entries: A7 $$0$$, B7 $$1$$, C7 $$2$$, D7 $$4$$, E7 $$-1$$, F7 $$-2$$, G7 $$-4$$.

The idea is to put entries in columns as follows: The only entries you really need enter are two rows of column A, and one of each of columns , H, O, R, T, and the rest is copying. Changing parameters only involves changing the data entered in the paragraph above. Changing the function only involves changing the data entry in column H and copying that into columns I through N and down $$100$$ rows. (The factors of $$\frac{1}{2}$$ in columns R, S, and T come from the fact that columns Q and R are convenient for copying, but are twice and four times the approximate derivatives.)

In A9, enter x
In B9, x+d
In C9, x+2d
In D9, x+4d
In E9, x-d
In F9, x-2d
In G9, x-4d
In H9, sin(x)
In I9, sin(x+d)
In J9, sin(x+2d)
In K9, sin(x+4d)
In L9, sin(x-d)
In M9, sin(x-2d)
In N9, sin(x-4d)
In O9, (sin(x+d)-sin(x-d))/(2d) which is the $$d$$ approximation to the derivative
In P9, (sin(x+2d)-sin(x-2d)/2d which is $$2$$ times the $$2d$$ approximation
In Q9, (sin(x+4d)-sin(x-4d)/2d which is $$4$$ times the $$4d$$ approximation
In R9, (4O-P/2)/3 which is the estimate with error proportional to $$d^4$$
In S9, (4P-Q/2)/3 which is $$2$$ times the estimate with error proportial to $$d^4$$
In T9, (16R-S/2)/15 which is the estimate with error proportional to $$d^6$$
In U9, A x data
In V9, H f(x) data
In W9, T f'(x) data
In X9, T-R accuracy check, if this number is small, error is small

Columns U,V, W and X are used for graphing our functions. If the maximum value in the X column is unacceptably large, $$d$$ should be reduced.

Here are the entries that need be entered. Suppose we start in row 10 (remember to have A7 =0, B7 =1, C7=2, D7=4, E7=-1, F7=-2, G7=-4).

A10 =$B$3+A$7*$B$6 A11 =A10+$C$5 Copy A11 down column A until B4 is obtained Copy A10 to B10, … G10, and A11 to B11, … G11 Copy B11 to G11 down those columns as far as you have copied column A H10 =sin(A10) copy across to I10, J10, K10, L10, M10, and N10 O10 =(I10-L10)/$B\$6/2 copy to P10 and Q10

R10 =(4*O10-P10/2)/3 copy to S10

T10 =(16*R10 - S10/2)/15

The following are repetitions of previous defined columns done to make scatter plots:

U10 =A10 which is $$x$$

V10 =H10 which is $$f(x)$$

W(10) =T10 which is the estimate of derivative of $$f(x)$$

X(10) =T10-R10 which is the improvement in estimate from using T instead of R

Now copy row 10 from column H through X down as far as column A goes.

Make an $$xy$$ scatter chart from the insert chart menu of the last 4 columns.

The parameters entered in B3-B6 can be changed there. The function can be changed in H10 and copied as above to I10 through N10 and down those columns.

If you have calculated the derivative of $$f$$ you can create a column for it as well and see if the plot is (or values are) any different from the numerical derivative.

Here is the result for the function $$\sin(x)$$ from $$x = 0$$ to $$x = 5$$.

Number of increments
Number of digits after decimal point

Exercises:

Set this up and apply it to the function $$\tan x$$ from $$x = 0$$ to $$x = 1.5$$, What happens if you make the upper limit $$1.6$$?