Differentiation:General Formulas Differentiation: Exponential and

Differentiation:General Formulas Differentiation: Exponential and

Differentiation:General Formulas
d
dx
d
dx
c = 0
fx + gx = f ′ x + g ′ x
d
dx
d
dx
d
dx
fxgx = fxg ′ x + gxf ′ x
d
dx
d
dx
fgx = f ′ gxg ′ x
d
dx
cfx = cf ′ x
fx − gx = f ′ x − g ′ x
fx
gx
=
gxf ′ x−fxg ′ x
gx 2
x n  = nx n−1
Differentiation: Exponential and Logarithmic Functions
d
dx
d
dx
e x  = e x
ln|x| =
a x  = a x ln a
d
dx
d
dx
1
x
log a x =
1
x lna
Differentiation: Trigonometric Functions
d
dx
d
dx
sin x = cos x
d
dx
d
dx
csc x = − csc x cot x
cos x = − sin x
sec x = sec x tan x
d
dx
d
dx
tan x = sec 2 x
cot x = − csc 2 x
Differentiation: Inverse Trigonometric Functions
d
dx
sin −1 x =
d
dx
csc −1 x = −
1
1−x 2
1
x x 2 −1
d
dx
cos −1 x = −
d
dx
sec −1 x =
1
1−x 2
1
x x 2 −1
d
dx
tan −1 x =
d
dx
1
cot −1 x = − 1+x
2
1
1+x 2
Integration Formulas
∫ udv = uv − ∫ vdu
∫ sec 2 u du = tan u + C
∫ csc u du = ln|csc u − cot u|+C
∫ csc 2 u du = − cot u + C
∫
∫ 1u du = ln|u| + C
∫ sec u tan u du = sec u + C
∫
∫ e u du = e u + C
∫ csc u cot u = − csc u + C
∫
∫ a u du =
∫ tan u du = ln|sec u|+C
∫
∫ sin u du = − cos u + C
∫ cot u du = ln|sin u|+C
∫
∫ cos u du = sin u + C
∫ sec u du = ln|sec u + tan u|+C
∫ u n du =
u n+1
n+1
au
lna
+ C,
n≠1
+C
1
a 2 −u 2
1
a 2 +u 2
1
du =
u u 2 −a 2
1
a 2 −u 2
1
u 2 −a 2
du = sin −1
1
a
du =
tan −1
1
a
Inverse Trig.
Algebraic
Trigonometric
u
a
sec −1
+C
u
a
+C
du =
1
2a
ln| u+a
u−a | + C
du =
1
2a
ln| u−a
u+a | + C
Integration by Parts (order for choosing u)
Logarithmic
+C
u
a
Exponential
Trigonometric Identities
1 + tan 2 θ = sec 2 θ
sinx + y = sin x cos y + cos x sin y
sin 2x = 2 sin x cos x
1 + cot 2 θ = csc 2 θ
sinx − y = sin x cos y − cos x sin y
cos 2x = cos 2 x − sin 2 x
tanx + y =
tanx − y =
tanx+tany
1−tanx tany
tanx−tany
1+tanx tany
cosx + y = cos x cos y − sin x sin y
= 2 cos 2 x − 1
cosx − y = cos x cos y + sin x sin y
= 1 − 2 sin 2 x
Miscellaneous
Quad. form.: x =
−b± b 2 −4ac
2a
2
Vol. of sphere: V =
4
3
πr 3
Eq. of circle: x − h + y − k 2 = r 2
Vol. of cylinder: V = πr 2 h
log a x = y  a y = x
x + y 2 = x 2 + 2xy + y 2
x 2 − y 2 = x + yx − y
x − y 2 = x 2 − 2xy + y 2
x 3 + y 3 = x + yx 2 − xy + y 2 
x + y 3 = x 3 + 3x 2 y + 3xy 2 + y 3
x 3 − y 3 = x − yx 2 + xy + y 2 
x − y 3 = x 3 − 3x 2 y + 3xy 2 − y 3
Unit Circle