Calculus Cheat Sheet

Trigonometric Formulas
1. sin 2 θ + cos 2 θ = 1 12.
2. 1 + tan 2 θ = sec 2 θ cos 2θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ
sin θ 1
3. 1 + cot 2 θ = csc 2 θ 13. tan θ = =
4. sin( −θ ) = − sin θ cosθ cot θ
cos(−θ ) = cosθ cosθ 1
5. 14. cot θ = =
6. tan(−θ ) = − tan θ sin θ tan θ
1
7. sin( A + B ) = sin A cos B + sin B cos A 15. secθ =
8. sin( A − B ) = sin A cos B − sin B cos A cosθ
1
9. 16. cscθ =
cos( A + B ) = cos A cos B − sin A sin B sin θ
10. π
17. cos( − θ ) = sin θ
cos( A − B) = cos A cos B + sin A sin B 2
11. sin 2θ = 2 sin θ cos θ π
18. sin( − θ ) = cos θ
2

Differentiation Formulas

d n d
1. ( x ) = nx n −1 10. (csc x ) = − csc x cot x
dx dx
d d x
2. ( fg ) = fg ′ + gf ′ 11. (e ) = e x
dx dx
d f gf ′ − fg ′ d x
3. ( )= 12. (a ) = a x ln a
dx g g2 dx
d d 1
4. f ( g ( x)) = f ′( g ( x )) g ′( x) 13. (ln x) =
dx dx x
d d 1
5. (sin x) = cos x 14. ( Arc sin x ) =
dx dx 1− x2
d d 1
6. (cos x) = − sin x 15. ( Arc tan x) =
dx dx 1+ x2
d d 1
7. (tan x ) = sec 2 x 16. ( Arc sec x) =
dx dx | x | x2 −1
d
8. (cot x ) = − csc 2 x dy dy du
dx 17. = × Chain Rule
dx dx dx
d
9. (sec x ) = sec x tan x
dx

Integration Formulas
1. ∫ a dx = ax + C
x n +1
∫ x dx = + C , n ≠ −1
n
2.
n +1
1
3. ∫ x dx = ln x + C
∫ e dx = e + C
x x
4.

ax
∫ a dx = +C
x
5.
ln a
6. ∫ ln x dx = x ln x − x + C
7. ∫ sin x dx = − cos x + C
8. ∫ cos x dx = sin x + C
9. ∫ tan x dx = ln sec x + C or − ln cos x + C
10. ∫ cot x dx = ln sin x + C
11. ∫ sec x dx = ln sec x + tan x + C
12. ∫ csc x dx = ln csc x − cot x + C
∫ sec x d x = tan x + C
2
13.

14. ∫ sec x tan x dx = sec x + C

∫ csc x dx = − cot x + C
2
15.

16. ∫ csc x cot x dx = − csc x + C

∫ tan x dx = tan x − x + C
2
17.
dx 1 x
18. ∫a 2
+x 2
= Arc tan  + C
a a
dx x
19. ∫ a2 − x2
= Arc sin   + C
a
dx 1 x 1 a
20. ∫x x2 − a2
=
a
Arc sec + C = Arc cos + C
a a x
 Formulas and Theorems
1a. Definition of Limit: Let f be a function defined on an open interval containing c (except
possibly at c ) and let L be a real number. Then
lim f ( x) = L means that for each ε > 0 there
x→a
exists a δ > 0 such that f ( x) − L < ε whenever 0 < x − c < δ .
1b. A function y = f (x) is continuous at x = a if
i). f(a) exists
ii). lim f ( x) exists
x→a
iii). lim = f (a)
x→a
2. Even and Odd Functions
1. A function y = f (x) is even if f ( − x) = f ( x) for every x in the function’s domain.
Every even function is symmetric about the y-axis.
2. A function y = f (x) is odd if f ( − x) = − f ( x) for every x in the function’s domain.
Every odd function is symmetric about the origin.
3. Periodicity
A function f (x) is periodic with period p ( p > 0) if f ( x + p ) = f ( x) for every value of x
.
2π
Note: The period of the function y = A sin( Bx + C ) or y = A cos( Bx + C ) is .
B
The amplitude is A . The period of y = tan x is π .
4. Intermediate-Value Theorem
A function y = f (x) that is continuous on a closed interval [ a, b] takes on every value between
f ( a) and f (b) .
Note: If f is continuous on [ a, b ] and f (a ) and f (b) differ in sign, then the equation
f ( x) = 0 has at least one solution in the open interval (a, b) .
5. Limits of Rational Functions as x → ±∞
f ( x)
i). lim = 0 if the degree of f ( x) < the degree of g ( x)
x → ±∞ g ( x )
x 2 − 2x
Example: lim =0
x → ∞ x3 + 3
f ( x)
ii). lim is infinite if the degrees of f ( x ) > the degree of g ( x)
x → ±∞ g ( x )

x3 + 2x
Example: lim =∞
x → ∞ x2 − 8
f ( x)
iii). lim is finite if the degree of f ( x) = the degree of g ( x)
x → ±∞ g ( x )

2 x 2 − 3x + 2 2
Example: lim =−
x → ∞ 10 x − 5 x 2 5
6. Horizontal and Vertical Asymptotes
1. A line y = b is a horizontal asymptote of the graph y = f (x) if either
lim f ( x) = b or lim f ( x) = b .
x→∞ x → −∞
2. A line x = a is a vertical asymptote of the graph y = f (x) if either
lim f ( x) = ±∞ or lim = ±∞
.
x → a+ x → a-

( x0 , y0 ) and ( x1, y1 ) are points on the graph of

7. Average and Instantaneous Rate of Change
i). Average Rate of Change: If
y = f (x) , then the average rate of change of y with respect to x over the interval
f ( x1 ) − f ( x 0 ) y1 − y 0 ∆y
[ x0 , x1 ] is = = .
x1 − x 0 x1 − x0 ∆x
ii). Instantaneous Rate of Change: If ( x 0 , y 0 ) is a point on the graph of y = f (x) , then
the instantaneous rate of change of y with respect to x at x 0 is f ′( x 0 ) .
f ( x + h) − f ( x )
8. f ′( x) = lim
h→0 h
9. The Number e as a limit
n
 1
i). lim 1 +  = e
n → +∞ n 
1
 n 
ii).
lim 1 +  n = e
n → 0 1 
10. Rolle’s Theorem
If f is continuous on [ a, b] and differentiable on ( a, b ) f (a) = f (b) , then there
such that
is at least one number c in the open interval ( a, b ) such that f ′(c) = 0 .
11. Mean Value Theorem
If f is continuous on [ a, b] and differentiable on ( a, b ) , then there is at least one number c
f (b) − f (a )
in ( a, b ) such that = f ′(c) .
b−a
12. Extreme-Value Theorem
If f is continuous on a closed interval [ a, b] , then f (x) has both a maximum and minimum
on [ a, b] .
13. To find the maximum and minimum values of a function y = f (x) , locate
1. the points where f ′(x) is zero or where f ′(x ) fails to exist.
2. the end points, if any, on the domain of f (x) .
Note: These are the only candidates for the value of x where f (x) may have a maximum or a
minimum.
14. Let f be differentiable for a < x < b and continuous for a a ≤ x ≤ b ,
1. If f ′( x) > 0 for every x in ( a, b ) , then f is increasing on [ a, b ] .
2. If f ′( x) < 0 for every x in ( a, b ) , then f is decreasing on [ a, b ] .