Successive Derivatives of $\frac{x}{1-x^{2}}$

Applying the Maple command

for n to 6 do diff(x/(1-x^2),x$n) od;

we obtain the successive derivatives of $\frac{x}{1-x^{2}}:$


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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$\allowbreak $
 

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This suggests that in general the n-th order derivative of $f(x)$ takes the form
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for some polynomial $p_{n+1}$ of degree $n+1.$ What is the relationship between $p_{n+1}$ and $p_{n+2}?$ To see this we make a new definition for the function $p(x).$ Taking the derivative
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we obtain$\allowbreak $
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Taking the Factor command, the expression becomes
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After combining powers and factoring, we obtain
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Thus the recursive formula consists of the initial condition
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together with
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The coefficients of $p_{n}$ 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


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Exercise

Consider the following functions:

  1. $x^{2}e^{1/x}$

  2. $\tan x$

  3. $e^{-1/x^{2}}$

  4. $e^{1/x}$

  5. $\sec x$

For each of the above function $f(x),$

  1. find the general form of the n-th derivative $f^{(n)}(x);$

  2. find the recursive formula associated with $f^{(n)}(x).$

  3. generate the coefficients occurred in $f^{(n)}(x)$ for $n=1,2,\cdots ,10$ with the spreadsheet.

Here is a part of the answers:

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)

 

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we see that number pattern associated with the successive derivatives of $e^{1/x}$ 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