Derivative Differentiation
1
1. ------
2
1 + x
x
2. - 2 ---------
2 2
(1 + x )
2
x 2
3 8 --------- - ---------
2 3 2 2
(1 + x ) (1 + x )
3
x x
4. - 48 --------- + 24 ---------
2 4 2 3
(1 + x ) (1 + x )
4 2
x x 24
5. 384 --------- - 288 --------- + ---------
2 5 2 4 2 3
(1 + x ) (1 + x ) (1 + x )
5 3
x x x
6. - 3840 --------- + 3840 --------- - 720 ---------
2 6 2 5 2 4
(1 + x ) (1 + x ) (1 + x )
6 4 2
x x x 720
7. 46080 --------- - 57600 --------- + 17280 --------- - ---------
2 7 2 6 2 5 2 4
(1 + x ) (1 + x ) (1 + x ) (1 + x )
7 5 3
x x x x
8. - 645120 --------- + 967680 --------- - 403200 --------- + 40320 ---------
2 8 2 7 2 6 2 5
(1 + x ) (1 + x ) (1 + x ) (1 + x )
9. 8 6 4 2
x x x x
10321920 --------- - 18063360 --------- + 9676800 --------- - 1612800 ---------
2 9 2 8 2 7 2 6
(1 + x ) (1 + x ) (1 + x ) (1 + x )
40320
+ ---------
2 5
(1 + x )
10. 9 7 5
x x x
- 185794560 ---------- + 371589120 --------- - 243855360 ---------
2 10 2 9 2 8
(1 + x ) (1 + x ) (1 + x )
3
x x
+ 58060800 --------- - 3628800 ---------
2 7 2 6
(1 + x ) (1 + x )
11. 10 8 6
x x x
3715891200 ---------- - 8360755200 ---------- + 6502809600 ---------
2 11 2 10 2 9
(1 + x ) (1 + x ) (1 + x )
4 2
x x 3628800
- 2032128000 --------- + 217728000 --------- - ---------
2 8 2 7 2 6
(1 + x ) (1 + x ) (1 + x )
12. 11 9 7
x x x
- 81749606400 ---------- + 204374016000 ---------- - 183936614400 ----------
2 12 2 11 2 10
(1 + x ) (1 + x ) (1 + x )
5 3
x x x
+ 71530905600 --------- - 11176704000 --------- + 479001600 ---------
2 9 2 8 2 7
(1 + x ) (1 + x ) (1 + x )
13. 12 10 8
x x x
1961990553600 ---------- - 5395474022400 ---------- + 5518098432000 ----------
2 13 2 12 2 11
(1 + x ) (1 + x ) (1 + x )
6 4
x x
- 2575112601600 ---------- + 536481792000 ---------
2 10 2 9
(1 + x ) (1 + x )
2
x 479001600
- 40236134400 --------- + ---------
2 8 2 7
(1 + x ) (1 + x )
14. 13 11
x x
- 51011754393600 ---------- + 153035263180800 ----------
2 14 2 13
(1 + x ) (1 + x )
9 7
x x
- 175352905728000 ---------- + 95647039488000 ----------
2 12 2 11
(1 + x ) (1 + x )
5 3
x x
- 25107347865600 ---------- + 2789705318400 ---------
2 10 2 9
(1 + x ) (1 + x )
x
- 87178291200 ---------
2 8
(1 + x )
15. 14 12
x x
1428329123020800 ---------- - 4642069649817600 ----------
2 15 2 14
(1 + x ) (1 + x )
10 8
x x
+ 5891857632460800 ---------- - 3682411020288000 ----------
2 13 2 12
(1 + x ) (1 + x )
6 4
x x
+ 1171676233728000 ---------- - 175751435059200 ----------
2 11 2 10
(1 + x ) (1 + x )
2
x 87178291200
+ 9763968614400 --------- - -----------
2 9 2 8
(1 + x ) (1 + x )
16. 15 13
x x
- 42849873690624000 ---------- + 149974557917184000 ----------
2 16 2 15
(1 + x ) (1 + x )
11 9
x x
- 208893134241792000 ---------- + 147296440811520000 ----------
2 14 2 13
(1 + x ) (1 + x )
7 5
x x
- 55236165304320000 ---------- + 10545086103552000 ----------
2 12 2 11
(1 + x ) (1 + x )
3
x x
- 878757175296000 ---------- + 20922789888000 ---------
2 10 2 9
(1 + x ) (1 + x )
17. 16 14
x x
1371195958099968000 ---------- - 5141984842874880000 ----------
2 17 2 16
(1 + x ) (1 + x )
12 10
x x
+ 7798677011693568000 ---------- - 6127531937759232000 ----------
2 15 2 14
(1 + x ) (1 + x )
8 6
x x
+ 2651335934607360000 ---------- - 618645051408384000 ----------
2 13 2 12
(1 + x ) (1 + x )
4 2
x x
+ 70300574023680000 ---------- - 3012881743872000 ----------
2 11 2 10
(1 + x ) (1 + x )
20922789888000
+ --------------
2 9
(1 + x )
18. 17 15
x x
- 46620662575398912000 ---------- + 186482650301595648000 ----------
2 18 2 17
(1 + x ) (1 + x )
13 11
x x
- 305948098151055360000 ---------- + 265155018397581312000 ----------
2 16 2 15
(1 + x ) (1 + x )
9 7
x x
- 130210053677383680000 ---------- + 36058168710660096000 ----------
2 14 2 13
(1 + x ) (1 + x )
5 3
x x
- 5258482936971264000 ---------- + 341459930972160000 ----------
2 12 2 11
(1 + x ) (1 + x )
x
- 6402373705728000 ----------
2 10
(1 + x )
19. 18 16
x x
1678343852714360832000 ---------- - 7132961374036033536000 ----------
2 19 2 18
(1 + x ) (1 + x )
14 12
x x
+ 12587578895357706240000 ---------- - 11931975827891159040000 ----------
2 17 2 16
(1 + x ) (1 + x )
10 8
x x
+ 6562586705340137472000 ---------- - 2109402869573615616000 ----------
2 15 2 14
(1 + x ) (1 + x )
6 4
x x
+ 378610771461931008000 ---------- - 33804533166243840000 ----------
2 13 2 12
(1 + x ) (1 + x )
2
x 6402373705728000
+ 1152427267031040000 ---------- - ----------------
2 11 2 10
(1 + x ) (1 + x )
20. 19 17
x x
- 63777066403145711616000 ---------- + 286996798814155702272000 ----------
2 20 2 19
(1 + x ) (1 + x )
15
x
- 542105064426738548736000 ----------
2 18
(1 + x )
13
x
+ 558049331027524976640000 ----------
2 17
(1 + x )
11
x
- 340061311094898032640000 ----------
2 16
(1 + x )
9 7
x x
+ 124689147401462611968000 ---------- - 26719103014599131136000 ----------
2 15 2 14
(1 + x ) (1 + x )
5 3
x x
+ 3082973424761438208000 ---------- - 160571532539658240000 ----------
2 13 2 12
(1 + x ) (1 + x )
x
+ 2432902008176640000 ----------
2 11
(1 + x )
See if you can work out the 21st derivative!
Last modified on Feb 21, 1998 by
Jason Hildebrand