This document contains important derivatives formulas for bachelor level mathematics. It includes the derivatives of common functions like sin, cos, tan, coth, and expressions involving constants, logarithms, and exponential functions. It also includes the derivatives of composite functions, Leibniz's theorem for computing derivatives of products, and taking higher order derivatives. The document was created by Atiq ur Rehman and is available online at http://www.mathcity.org.

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Merging man and maths
Some Important Derivative
For Bachelor Level
Available online @ http://www.mathcity.org, Version: 1.1.2
0
d
c
dx
· = where c is constant 1n nd
x nx
dx
-
· =
sin cos
cos sin
d
x x
dx
d
x x
dx
ì
ïï
í
ï
ïî
· =
· =-
2
2
tan sec
cot csc
d
x x
dx
d
x x
dx
· =
· =-
csc csc cot
sec sec tan
d
x x x
dx
d
x x x
dx
· =-
· =
1
2
1
2
1
1
1
1
d
Sin x
dx x
d
Cos x
dx x
-
-
ì
ï
ï
í
ï
ï
î
· =
-
-
· =
-
1
2
1
2
1
1
1
1
d
Tan x
dx x
d
Cot x
dx x
-
-
· =
+
-
· =
+
1
2
1
2
1
1
1
1
d
Sec x
dx x x
d
Csc x
dx x x
-
-
· =
-
-
· =
-
lnx x
x x
d
a a a
dx
d
e e
dx
ì
ïï
í
ï
ïî
· =
· =
1
log
ln
1
ln
a
d
x
dx x a
d
x
dx x
· =
· =
sinh cosh
cosh sinh
d
x x
dx
d
x x
dx
ì
ïï
í
ï
ïî
· =
· =
2
2
tanh sech
coth csch
d
x x
dx
d
x x
dx
· =
· = -
sech sech tanh
csch csch coth
d
x x x
dx
d
x x x
dx
· =-
· =-
1
2
1
2
1
1
1
1
d
Sinh x
dx x
d
Cosh x
dx x
-
-
ì
ï
ï
í
ï
ï
î
· =
+
· =
-
1
2
1
2
1
1
1
1
d
Tanh x
dx x
d
Coth x
dx x
-
-
· =
-
· =
-
1
2
1
2
1
1
1
1
d
Sech x
dx x x
d
Csch x
dx x x
-
-
-
· =
-
-
· =
+
Some Standard nth Derivative
·
( )
( )!
( )
!
n m nm n
n
d m
ax b a ax b if m n
dx m n
-
+ = + ³
-
·
( )
1
1 !1
( )
n nn
n n
n ad
dx ax b ax b +
æ ö
ç ÷
è ø
-
=
+ +
·
( )1
( 1) 1 !
ln( )
( )
n nn
n n
n ad
ax b
dx ax b
-
é ùë û
- -
+ =
+
·
n
ax n ax
n
d
e a e
dx
=
· sin( ) sin .
2
n
n
n
d
ax b a ax b n
dx
pæ ö
ç ÷
è ø
+ = + + · cos( ) cos .
2
n
n
n
d
ax b a ax b n
dx
pæ ö
ç ÷
è ø
+ = + +
· 2 2 12sin( ) ( ) sin tan
nn
ax ax
n
d b
e bx c a b e bx c n
dx a
-æ ö
ç ÷
è ø
+ = + + +
· 2 2 12cos( ) ( ) cos tan
nn
ax ax
n
d b
e bx c a b e bx c n
dx a
-æ ö
ç ÷
è ø
+ = + + +
Leibniz’s Theorem
( ) ( 1) ( 2) 1
0 1 2 1( )
n
n n n n n n n n n n
nnn
d
uv C u v C u v C u v C u v C uv
dx
- - -
-¢ ¢¢ ¢= + + +××××××××××××××+ +
Made by Atiq ur Rehman ( atiq@mathcity.org )
http://www.mathcity.org
Submit errors at http://www.mathcity.org/error