Basic Differentiation Formulas
In the table=
below, u f=
( x) and v g ( x) represent differentiable functions of x
d
(c ) = 0
dx
2. Derivative of a constant multiple:
d
[c ⋅ f ( x)] =
c ⋅ f ′( x)
dx
1. Derivative of a constant:
3. Derivative of a sum or difference:
d
du dv
[u ± v] =
±
dx
dx dx
4. Product Rule:
d
dv
du
[u ⋅ v] = u ⋅ + v ⋅
dx
dx
dx
d u 
5. Quotient Rule:
=
dx  v 
6. Chain Rule:
v
du
dv
−u
dx
dx
v2
d
du
[ f=
(u )] f ′(u ) ⋅
dx
dx
General Formulas:
With the chain rule . . .
d n
x = nx n −1
dx
d x
e = ex
dx
d x
a = (ln a )a x
dx
d
1
(ln x) =
dx
x
d
(sin x) = cos x
dx
d
(cos x) = − sin x
dx
d
(tan x) = sec 2 x
dx
d
(cot x) = − csc 2 x
dx
d
(sec x) = sec x tan x
dx
d
(csc x) = − csc x cot x
dx
d
1
(sin −1 x) =
dx
1 − x2
d
1
(tan −1 x) =
dx
1 + x2
d n
du
u = nu n −1
dx
dx
d u
du
e = eu
dx
dx
d u
du
a = (ln a )a u
dx
dx
d
1 du
(ln u ) =
dx
u dx
d
du
(sin u ) = cos u
dx
dx
d
du
(cos u ) = − sin u
dx
dx
d
du
(tan u ) = sec 2 u
dx
dx
d
du
(cot u ) = − csc 2 u
dx
dx
d
du
(sec u ) = sec u tan u
dx
dx
d
du
(csc u ) = − csc u cot u
dx
dx
d
1
(sin −1 u ) =
du
dx
1− u2
d
1
(tan −1 u ) =
du
dx
1+ u2
Table of Integration Formulas
u n +1
+ C , n ≠ −1
n +1
1.
n
du
∫u =
2.
du ln | u | +C
∫=
u
3.
∫ e du= e + C
4.
a du
a +C
∫=
ln a
5.
− cos u + C
∫ sin udu =
6.
= sin u + C
∫ cos udu
7.
= tan u + C
∫ sec udu
8.
− cot u + C
∫ csc udu =
9.
= sec u + C
∫ tan u sec udu
10.
− csc u + C
∫ cot u csc udu =
11.
∫ tan udu= ln | sec u | +C or − ln | cos u | +C
12.
cot udu ln | sin u | +C
∫=
13.
∫ sec udu= ln | sec u + tan u | +C
14.
− ln | csc u + cot u | +C
∫ csc udu =
15.
du
arctan + C
∫ a=
+u
a
a
16.
du arcsin + C
∫ a −=
a
u
1
u
u
1
u
u
2
2
1
2
1
u
2
1
2
u
2