Differen a on Rules
1.
d
(cx) = c
dx
10.
d x
(a ) = ln a · ax
dx
19.
d ( −1 )
1
sin x = √
dx
1 − x2
28.
d
(sech x) = − sech x tanh x
dx
2.
d
(u ± v) = u′ ± v′
dx
11.
d
1
(ln x) =
dx
x
20.
d ( −1 )
−1
cos x = √
dx
1 − x2
29.
d
(csch x) = − csch x coth x
dx
12.
d
1
1
(loga x) =
·
dx
ln a x
21.
−1
d ( −1 )
csc x = √
dx
|x| x2 − 1
30.
d
(coth x) = − csch2 x
dx
13.
d
(sin x) = cos x
dx
22.
d ( −1 )
1
sec x = √
dx
|x| x2 − 1
31.
)
d (
1
cosh−1 x = √
dx
x2 − 1
d
(u · v) = uv′ + u′ v
dx
( )
d u
vu′ − uv′
4.
=
dx v
v2
3.
5.
d
(u(v)) = u′ (v)v′
dx
14.
d
(cos x) = − sin x
dx
23.
d ( −1 )
1
tan x =
dx
1 + x2
32.
)
d (
1
sinh−1 x = √
dx
x2 + 1
6.
d
(c) = 0
dx
15.
d
(csc x) = − csc x cot x
dx
24.
d ( −1 )
−1
cot x =
dx
1 + x2
33.
)
d (
−1
sech−1 x = √
dx
x 1 − x2
7.
d
(x) = 1
dx
16.
d
(sec x) = sec x tan x
dx
25.
d
(cosh x) = sinh x
dx
34.
)
−1
d (
csch−1 x = √
dx
|x| 1 + x2
8.
d n
(x ) = nxn−1
dx
17.
d
(tan x) = sec2 x
dx
26.
d
(sinh x) = cosh x
dx
35.
)
1
d (
tanh−1 x =
dx
1 − x2
9.
d x
(e ) = ex
dx
18.
d
(cot x) = − csc2 x
dx
27.
d
(tanh x) = sech2 x
dx
36.
)
1
d (
coth−1 x =
dx
1 − x2
Integra on Rules
∫
∫
c · f(x) dx = c
1.
∫
f(x) dx
∫
∫
g(x) dx
14.
15.
∫
1 dx = x + C
∫
xn dx =
∫
1 n+1
x
+ C, n ̸= −1
n+1
ex dx = ex + C
6.
16.
ln x dx = x ln x − x + C
∫
ax dx =
8.
∫
9.
∫
10.
1
· ax + C
ln a
1
dx = ln |x| + C
x
cos x dx = sin x + C
∫
11.
27.
csc2 x dx = − cot x + C
28.
sec x tan x dx = sec x + C
29.
sin x dx = − cos x + C
csc x cot x dx = − csc x + C
∫
cos2 x dx =
20.
∫
( )
1
1
x + sin 2x + C
2
4
( )
1
1
x − sin 2x + C
2
4
( )
∫
1
1
x
−1
22.
dx
=
tan
+C
x2 + a 2
a
a
21.
sin2 x dx =
dx = sin−1
sinh x dx = cosh x + C
tanh x dx = ln(cosh x) + C
coth x dx = ln | sinh x| + C
∫
∫
19.
x2
∫
∫
18.
1
a2
∫
∫
17.
√
∫
∫
∫
7.
sec2 x dx = tan x + C
∫
23.
∫
0 dx = C
5.
26.
∫
∫
4.
cot x dx = ln | sin x| + C
sec x dx = ln | sec x + tan x| + C
13.
∫
f(x) dx ±
3.
csc x dx = − ln | csc x + cot x| + C
∫
f(x) ± g(x) dx =
2.
( )
x
+C
a
−
( )
∫
1
1
|x|
√
24.
dx = sec−1
+C
a
a
x x2 − a 2
∫
25.
cosh x dx = sinh x + C
tan x dx = − ln | cos x| + C
12.
√
∫
√
30.
∫
1
x2
−
a2
1
x2 + a 2
dx = ln x +
√
x2 − a 2 + C
dx = ln x +
√
x2 + a 2 + C
a+x
1
1
dx =
ln
+C
a 2 − x2
2a
a−x
)
(
∫
1
1
x
√
√
32.
dx = ln
+C
a
x a 2 − x2
a + a 2 − x2
∫
1
1
x
√
√
33.
dx = ln
+C
a
x x2 + a 2
a + x2 + a 2
31.
The Unit Circle
Defini ons of the Trigonometric Func ons
y
Unit Circle Defini on
(
√ )
−1, 3
( √ √ 2) 2
− 22 , 22
2π/3
)
( √
− 23 , 12
3π/4
120◦
◦
5π/6 135
(
(0, 1)
π/2
90◦
π
60◦
2
√
, 22
(√
3
2
)
,
1
2
(x, y)
sin θ = y
)
y
θ
π/6
30◦
0◦
330◦
11π/6
315◦
270◦
3π/2
(0, −1)
300◦
(√
3
)
e
us
n
te
po
y
H
Pythagorean Iden
2
2
sin x + cos x = 1
2
2
tan x + 1 = sec x
1 + cot2 x = csc2 x
es
1
x
tan θ =
y
x
cot θ =
x
y
sin θ =
O
H
csc θ =
H
O
cos θ =
A
H
sec θ =
H
A
tan θ =
O
A
cot θ =
A
O
θ
Adjacent
Common Trigonometric Iden
sec θ =
Right Triangle Defini on
1
2 ,−2
(√
5π/3
√ )
2
2
2 ,− 2
(
√ )
3
1
2,− 2
7π/4
1
y
x
(1, 0)
0
cos θ = x
csc θ =
x
x
Opposite
( √
− 23 , − 2
( √
4π/3
√ )
− 22 , − 22
)
(
√
− 12 , − 23
(√
π/4
45◦
y
2
2
180◦
210◦
7π/6
225◦
)
◦
1
5π/4 240
3
)
π/3
150◦
(−1, 0)
1
2,
√
es
Cofunc on Iden es
)
(π
− x = cos x
sin
2
(π
)
cos
− x = sin x
2
(π
)
tan
− x = cot x
2
Sum to Product Formulas
(
)
(
)
x+y
x−y
sin x + sin y = 2 sin
cos
2
2
)
(
)
(
x+y
x−y
cos
sin x − sin y = 2 sin
2
2
(
)
(
)
x+y
x−y
cos x + cos y = 2 cos
cos
2
2
(
) (
)
x+y
x−y
cos x − cos y = −2 sin
sin
2
2
Product to Sum Formulas
)
1(
sin x sin y =
cos(x − y) − cos(x + y)
2
)
1(
cos x cos y =
cos(x − y) + cos(x + y)
2
)
1(
sin x cos y =
sin(x + y) + sin(x − y)
2
(π
)
− x = sec x
)
sec
− x = csc x
2
(π
)
cot
− x = tan x
2
csc
2
(π
Double Angle Formulas
sin 2x = 2 sin x cos x
cos 2x = cos2 x − sin2 x
= 2 cos2 x − 1
= 1 − 2 sin2 x
2 tan x
tan 2x =
1 − tan2 x
Power–Reducing Formulas
1 − cos 2x
sin2 x =
2
1 + cos 2x
2
cos x =
2
1
−
cos
2x
tan2 x =
1 + cos 2x
Even/Odd Iden
es
sin(−x) = − sin x
cos(−x) = cos x
tan(−x) = − tan x
csc(−x) = − csc x
sec(−x) = sec x
cot(−x) = − cot x
Angle Sum/Difference Formulas
sin(x ± y) = sin x cos y ± cos x sin y
cos(x ± y) = cos x cos y ∓ sin x sin y
tan x ± tan y
tan(x ± y) =
1 ∓ tan x tan y
Areas and Volumes
Triangles
Right Circular Cone
h = a sin θ
c
a
h
Area = 12 bh
θ
Law of Cosines:
c2 = a2 + b2 − 2ab cos θ
b
Parallelograms
Volume = 13 πr2 h
Surface Area =
√
πr r2 + h2 + πr2
h
r
Right Circular Cylinder
Area = bh
r
Volume = πr2 h
h
Surface Area =
Trapezoids
Area =
1
2 (a
h
2πrh + 2πr2
b
Sphere
a
+ b)h
Volume = 43 πr3
r
Surface Area =4πr2
h
b
Circles
General Cone
Area = πr2
Area of Base = A
r
Circumference = 2πr
h
Volume = 13 Ah
A
Sectors of Circles
General Right Cylinder
θ in radians
Area =
s = rθ
1 2
2 θr
s
Area of Base = A
Volume = Ah
θ
r
h
A
Algebra
Factors and Zeros of Polynomials
Let p(x) = an xn + an−1 xn−1 + · · · + a1 x + a0 be a polynomial. If p(a) = 0, then a is a zero of the polynomial and a solu on of
the equa on p(x) = 0. Furthermore, (x − a) is a factor of the polynomial.
Fundamental Theorem of Algebra
An nth degree polynomial has n (not necessarily dis nct) zeros. Although all of these zeros may be imaginary, a real
polynomial of odd degree must have at least one real zero.
Quadra c Formula
If p(x) = ax2 + bx + c, and 0 ≤ b2 − 4ac, then the real zeros of p are x = (−b ±
√
b2 − 4ac)/2a
Special Factors
x3 − a3 = (x − a)(x2 + ax + a2 )
x2 − a2 = (x − a)(x + a)
3
3
2
2
x + a = (x + a)(x − ax + a )
x4 − a4 = (x2 − a2 )(x2 + a2 )
n(n−1)
(x + y)n = xn + nxn−1 y + 2! xn−2 y2 + · · · + nxyn−1 + yn
n−2 2
(x − y)n = xn − nxn−1 y + n(n−1)
y − · · · ± nxyn−1 ∓ yn
2! x
Binomial
Theorem
2
2
(x − y)2 = x2 − 2xy + y2
(x − y)3 = x3 − 3x2 y + 3xy2 − y3
(x − y)4 = x4 − 4x3 y + 6x2 y2 − 4xy3 + y4
(x + y) = x + 2xy + y2
(x + y)3 = x3 + 3x2 y + 3xy2 + y3
(x + y)4 = x4 + 4x3 y + 6x2 y2 + 4xy3 + y4
Ra onal Zero Theorem
If p(x) = an xn + an−1 xn−1 + · · · + a1 x + a0 has integer coefficients, then every rational zero of p is of the form x = r/s,
where r is a factor of a0 and s is a factor of an .
Factoring by Grouping
acx3 + adx2 + bcx + bd = ax2 (cs + d) + b(cx + d) = (ax2 + b)(cx + d)
Arithme c Opera ons
ab + ac = a(b + c)
(a)
( bc ) =
(a) (d)
b
c
=
a
c
ad + bc
+ =
b d
bd
(a)
ad
bc
b
c
=
a b
a+b
= +
c
c
c
a
bc
a
ac
( )=
b
b
c
d
a
( )
b
ab
=
c
c
a−b
b−a
=
c−d
d−c
ab + ac
=b+c
a
Exponents and Radicals
a0 = 1, a ̸= 0
(ab)x = ax bx
( a )x
√
n
b
ax
= x
b
am
=a
m/n
ax ay = ax+y
a
−x
1
= x
a
√
√
n
a = a1/2
√ √
ab = n a n b
ax
= ax−y
ay
√
n
a = a1/n
√
x y
(a ) = a
xy
n
√
n
a
a
=√
n
b
b
Addi onal Formulas
Summa
on Formulas:
n
∑
n
∑
n(n + 1)
2
i=1
)2
(
n
∑
n(n + 1)
3
i =
2
i=1
c = cn
i=1
n
∑
i2 =
i=1
n(n + 1)(2n + 1)
6
i=
Trapezoidal
Rule:
∫
b
]
∆x [
f(x1 ) + 2f(x2 ) + 2f(x3 ) + ... + 2f(xn ) + f(xn+1 )
2
f(x) dx ≈
a
with Error ≤
]
(b − a)3 [
max f ′′ (x)
2
12n
Simpson’s
Rule:
∫
b
]
∆x [
f(x1 ) + 4f(x2 ) + 2f(x3 ) + 4f(x4 ) + ... + 2f(xn−1 ) + 4f(xn ) + f(xn+1 )
3
f(x) dx ≈
a
with Error ≤
]
(b − a)5 [
max f (4) (x)
4
180n
Arc Length:
∫
b
L=
Surface of Revolu on:
∫
√
1 + f ′ (x)2 dx
b
S = 2π
√
f(x) 1 + f ′ (x)2 dx
a
a
(where f(x) ≥ 0)
∫
S = 2π
b
x
√
1 + f ′ (x)2 dx
a
(where a, b ≥ 0)
Work Done by a Variable Force:
∫
W=
Force Exerted by a Fluid:
∫
b
F(x) dx
F=
a
b
w d(y) ℓ(y) dy
a
Taylor Series Expansion for f(x):
pn (x) = f(c) + f ′ (c)(x − c) +
f ′′′ (c)
f (n) (c)
f ′′ (c)
(x − c)2 +
(x − c)3 + ... +
(x − c)n
2!
3!
n!
Maclaurin Series Expansion for f(x), where c = 0:
pn (x) = f(0) + f ′ (0)x +
f ′′ (0) 2 f ′′′ (0) 3
f (n) (0) n
x +
x + ... +
x
2!
3!
n!
Summary of Tests for Series:
Test
Series
∞
∑
nth-Term
n=1
∞
∑
Geometric Series
Condi on(s) of
Convergence
an
Condi on(s) of
Divergence
Comment
lim an ̸= 0
This test cannot be used to
show convergence.
|r| ≥ 1
1
1−r
( a
)
∑
Sum =
bn − L
n→∞
|r| < 1
rn
n=0
Telescoping Series
p-Series
∞
∑
n=1
Integral Test
(bn − bn+a )
n=1
∞
∑
lim bn = L
n→∞
1
(an + b)p
∞
∑
an
an
n=0
Limit Comparison
∞
∑
an
n=0
Ra o Test
∞
∑
an
n=0
∞
a(n) dn
a(n) dn
1
1
∞
∑
Direct Comparison
p≤1
∫
∞
is convergent
n=0
∞
∑
n=1
p>1
∫
Sum =
is divergent
∞
∑
bn
n=0
an = a(n) must be
con nuous
bn
n=0
converges and
diverges and
0 ≤ a n ≤ bn
∞
∑
bn
0 ≤ bn ≤ a n
∞
∑
bn
n=0
n=0
converges and
diverges and
lim an /bn ≥ 0
lim an /bn > 0
n→∞
n→∞
an+1
lim
<1
n→∞ an
an+1
lim
>1
n→∞ an
Also diverges if
lim an /bn = ∞
n→∞
{an } must be posi ve
Also diverges if
lim an+1 /an = ∞
n→∞
Root Test
∞
∑
n=0
an
( )1/n
lim an
<1
n→∞
( )1/n
lim an
>1
n→∞
{an } must be posi ve
Also diverges if
( )1/n
lim an
=∞
n→∞