SOME FORMULAE ARE NICE TO KNOW AND SOME ARE NEED TO KNOW-SHARING BOTH
Calculus Formulas
n +1
∫
d n
x
x = nx n −1 and x n dx =
+c
dx
n +1
d ⎡ f (x ) ⎤ g (x ) ⋅ f ' (x ) − f (x ) ⋅ g ' (x )
Quotient Rule:
⎢
⎥=
dx ⎣ g (x ) ⎦
[g (x )]2
Power Rules:
Chain Rule:
Product Rule:
d
[ f (x ) ⋅ g (x )] = f (x ) ⋅ g ' (x ) + f ' (x ) ⋅ g (x )
dx
Reciprocal Rule:
d ⎡ 1 ⎤ − g ' (x )
⎢
⎥=
dx ⎣ g (x ) ⎦ [g (x )]2
d
( f o g )(x ) = f ' [g (x )] ⋅ g ' (x )
dx
Trigonometric Functions
Derivative
Integral
d
sin x = cos x
dx
∫ sin x dx = − cos x + c
d
cos x = − sin x
dx
∫ cos x dx = sin x + c
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
∫ tan x dx = ln sec x + c
∫ sec x dx = tan x + c
∫ cot x dx = ln sin x + c
∫ csc x dx = − cot x + c
∫ sec x dx = ln sec x + tan x + c
∫ sec x ⋅ tan x dx = sec x + c
∫ csc x dx = ln csc x − cot x + c
∫ csc x ⋅ cot x dx = − csc x + c
2
2
⎧ 2
2
⎪sin x + cos x = 1
⎪
⎪
Identities: ⎨1 + cot 2 x = csc 2 x
⎪
⎪tan 2 x + 1 = sec 2 x
⎪
⎩
sin 2 x = 2 sin x cos x
cos 2 x = cos 2 x − sin 2 x
cos(x + y ) = cos x cos y − sin x sin y
Exponential Functions
Derivative
Integral
( )
∫ e dx = e + c
( )
∫
d x
e = ex
dx
d x
b = (ln b )b x
dx
x
x
b x dx =
bx
+c
ln b
Definition of Log base b: log b N = x ⇔ b x = N
( )
( )
⎧⎪ln e x = x
Identities: ⎨
⎪⎩log b b x = x
e ln x = x
b
log b x
=x
1 + cos 2 x
2
1
cos
2x
−
sin 2 x =
2
sin (x + y ) = sin x cos y + cos x sin y
cos 2 x =
Logarithmic Functions
Derivative
d
(ln x ) = 1
dx
x
d
(log b x ) = ( 1 )
dx
ln b x
Integral
∫ x dx = ln x + c
1
Change of Base Formula: log b x =
ln e = log 10 = log b b = 1
ln 1 = log 1 = log b 1 = 0
ln x log x
=
ln b log b