differentiation_formulas.pdf
1
Differentiation Formulas This page contains a list of commonly used differentiation formulas. x 1. Derivative of a Constant (k): [] 0 d k dx = 2. Constant-Multiple Rule: [ ] () '( ) d k fx k f x dx ⋅ = ⋅ 3. Sum/Difference Rule: [ () ( )] '( ) '( ) d f x gx f x g x dx ± = ± 4. Power Rule: 1 r r d x rx dx − ⎡ ⎤= ⎣ ⎦ 5. Product Rule: [ () ( )] () '( ) ( ) '( ) d f xgx fxg x gx f x dx = + 6. Quotient Rule: [ ] 2 () '( ) () '( ) () () () d fx f xgx g x fx dx gx gx ⎡ ⎤ − = ⎢ ⎥ ⎣ ⎦ 7. Reciprocal Rule: 2 1 ' () [ ( )] d g dx gx gx ⎡ ⎤ − = ⎢ ⎥ ⎣ ⎦ () x 8. Chain Rule: ( ) ( ) () ' () '( ) d f gx f gx g x dx = 9. x d e e dx ⎡ ⎤ = ⎣ ⎦ 10. [ ] 1 ln( ) ( 0) d x x dx x = > 11. ln x x d a a a dx ⎡ ⎤= ⋅ ⎣ ⎦ 12. [ ] 1 log ln a d x dx x a = 13. [ ] sin cos d x x dx = 14. [ ] cos sin d x x dx =− 15. [ ] 2 tan sec d x x dx = 16. [ ] sec sec tan d x x x dx = 17. [ ] csc csc cot d x x x dx =− 18. [ ] 2 cot csc d x x dx =− 19. 1 2 1 sin 1 d x dx − ⎡ ⎤= ⎣ ⎦ − x 20. 1 2 1 tan 1 d x dx x − ⎡ ⎤= ⎣ ⎦ + 21. 1 2 1 sec 1 d x dx x x − ⎡ ⎤= ⎣ ⎦ − 22. 1 2 1 cos 1 d x dx − ⎡ ⎤=− ⎣ ⎦ − x 23. 1 2 1 cot 1 d x dx x − ⎡ ⎤=− ⎣ ⎦ + 24. 1 csc d x dx − ⎡ ⎤=− ⎣ ⎦ 2 1 1 x x − revised 12/07
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Transcript of differentiation_formulas.pdf
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Differentiation Formulas This page contains a list of commonly used differentiation formulas. x1. Derivative of a Constant (k):
[ ] 0d kdx
= 2. Constant-Multiple Rule:
[ ]( ) '( )d k f x k f xdx
= 3. Sum/Difference Rule:
[ ( ) ( )] '( ) '( )d f x g x f x g xdx
= 4. Power Rule:
1r rd x rxdx
= 5. Product Rule:
[ ( ) ( )] ( ) '( ) ( ) '( )d f x g x f x g x g x f xdx
= + 6. Quotient Rule:
[ ]2( ) '( ) ( ) '( ) ( )( ) ( )
d f x f x g x g x f xdx g x g x
= 7. Reciprocal Rule:
21 ' ( ) [ ( )]
d gdx g x g x
= ( )x
8. Chain Rule:
( ) ( )( ) ' ( ) '( )d f g x f g x g xdx
=
9. xd e edx
=
10. [ ] 1ln( ) ( 0)d x xdx x
= >
11. lnx xd a a adx
=
12. [ ] 1loglna
d xdx x a
=
13. [ ]sin cosd x xdx
=
14. [ ]cos sind x xdx
=
15. [ ] 2tan secd x xdx
=
16. [ ]sec sec tand x x xdx
=
17. [ ]csc csc cotd x x xdx
=
18. [ ] 2cot cscd x xdx
=
19. 12
1sin1
d xdx
= x20. 1 2
1tan1
d xdx x
= +21. 1
2
1sec1
d xdx x x
=
22. 12
1cos1
d xdx
= x23. 1 2
1cot1
d xdx x
= +
24. 1cscd xdx
= 21
1x x
revised 12/07
