# Successive Derivatives of

Applying the Maple command

for n to 6 do diff(x/(1-x^2),x$n) od; we obtain the successive derivatives of The coefficients are listed in this table:  1 1 0 2 0 0 0 0 0 0 0 6 0 8 0 0 0 0 0 6 0 48 0 48 0 0 0 0 0 120 0 480 0 384 0 0 0 120 0 2160 0 5760 0 3840 0 0 0 5040 0 40320 0 80640 0 46080 0 5040 0 161280 0 806400 0 1290240 0 645120 Let us apply the Maple command for n to 6 do simplify(diff(x/(1-x^2),x$n)) od;

we obtain

This suggests that in general the n-th order derivative of takes the form

for some polynomial of degree What is the relationship between and To see this we make a new definition for the function Taking the derivative

we obtain

Taking the Factor command, the expression becomes

After combining powers and factoring, we obtain

Thus the recursive formula consists of the initial condition

together with

The coefficients of are listed with the spreadsheet:

 1 1 0 1 0 0 0 0 0 0 0 3 0 1 0 0 0 0 0 1 0 6 0 1 0 0 0 0 0 5 0 10 0 1 0 0 0 1 0 15 0 15 0 1 0 0 0 7 0 35 0 21 0 1 0 1 0 28 0 70 0 28 0 1

# Exercise

Consider the following functions:

For each of the above function

1. find the general form of the n-th derivative

2. find the recursive formula associated with

3. generate the coefficients occurred in for with the spreadsheet.

Here is a part of the answers:

• Coefficients associated with

 1 1 1 0 2 0 2 0 0 0 0 0 2 0 8 0 6 0 0 0 0 0 16 0 40 0 24 0 0 0 16 0 136 0 240 0 120 0 0 0 272 0 1232 0 1680 0 720 0 272 0 3968 0 12096 0 13440 0 5040

• For the n-th order derivative of takes the form

for some polynomial of degree The recursive relationship of is given by

Accordingly, the coefficients of the polynomials can be generated by the spreadsheet:

 -1 1 4 0 0 0 0 0 0 -1 -10 -20 0 0 0 0 0 1 18 90 120 0 0 0 0 -1 -28 -252 -840 -840 0 0 0 1 40 560 3360 8400 6720 0 0 -1 -54 -1080 -10080 -45360 -90720 -60480 0 1 70 1890 25200 176400 635040 1058400 604800

The result can be checked with a simple Maple command

p:=1;for n from 3 to 7 do p:=expand(-(1+(2*n-2)*x)*p+diff(p,x)*x^2)

• To investigate the number pattern associated with write

The expansion of the successive derives are then expressed as a linear combinations of From

it is clear that in general we have

The associated number pattern is generated based on this principle.

 1 0 1 0 0 0 0 0 0 0 1 0 2 0 0 0 0 0 0 0 5 0 6 0 0 0 0 0 5 0 28 0 24 0 0 0 0 0 61 0 180 0 120 0 0 0 61 0 662 0 1320 0 720 0 0 0 1385 0 7266 0 10920 0 5040 0 1385 0 24568 0 83664 0 100800 0 40320

• The number pattern associated with the successive derivatives of of the form

is

 -1 2 1 0 0 0 0 0 0 -6 -6 -1 0 0 0 0 0 24 36 12 1 0 0 0 0 -120 -240 -120 -20 -1 0 0 0 720 1800 1200 300 30 1 0 0 -5040 -15120 -12600 -4200 -630 -42 -1 0 40320 141120 141120 58800 11760 1176 56 1

• From

we see that number pattern associated with the successive derivatives of of the form

is

 -1 1 2 0 0 0 0 0 0 -1 -6 -6 0 0 0 0 0 1 12 36 24 0 0 0 0 -1 -20 -120 -240 -120 0 0 0 1 30 300 1200 1800 720 0 0 -1 -42 -630 -4200 -12600 -15120 -5040 0 1 56 1176 11760 58800 141120 141120 40320

• The number pattern associated with the successive derivatives of is

 2 -6 4 0 0 0 0 0 24 -36 8 0 0 0 0 -120 300 -144 16 0 0 0 720 -2640 2040 -480 32 0 0 -5040 25200 -27720 10320 -1440 64 0 40320 -262080 383040 -199920 43680 -4032 128