1
Rules and Formulas
Definition of the derivative:
f (x + h)
h!0
h
f 0 (x) = lim
f (x)
Table of basic derivatives:
f (x)
C
xn
f 0 (x)
0
nxn
1
ex
sin x
ex
cos x
cos x
tan x
sin x sec2 x
ln |x|
1
x
sin
1
x tan
p 1
1 x2
1
1+x2
Di↵erentiation Rules:
Power rule:
Constant multiple rule:
Sum/di↵erence rule:
Product rule:
Quotient rule:
Chain rule:
d n
[x ] = nxn 1
dx
d
[c · f (x)] = c · f 0 (x)
dx
d
[f (x) ± g(x)] = f 0 (x) ± g 0 (x)
dx
d
[f (x) · g(x)] = f (x) · g 0 (x) + g(x) · f 0 (x)
dx 
d f (x)
g(x) · f 0 (x) f (x) · g 0 (x)
=
dx g(x)
[g(x)]2
d
[f (g(x))] = f 0 (g(x)) · g 0 (x)
dx
L’Hospital’s Rule: Suppose lim f (x) = 0 and lim g(x) = 0, or
x!a
lim f (x) = ±1 and lim g(x) = ±1. Then
x!a
x!a
x!a
f (x)
f 0 (x)
lim
= lim 0
x!a g(x)
x!a g (x)
as long as the limit on the right exists (or is ±1).
1
x
2
Theorems
Intermediate Value Theorem: If f is continuous on [a, b] and N is a number
between f (a) and f (b), then there is a number c in (a, b) such that f (c) = N .
Extreme Value Theorem: If f is continuous on [a, b], then it has an absolute
max and an absolute min on [a, b].
Rolle’s Theorem: If f is continuous on [a, b], di↵erentiable on (a, b), and
f (a) = f (b), then there is a number c in (a, b) such that f 0 (c) = 0.
Mean Value Theorem: If f is continuous on [a, b] and di↵erentiable on
(a, b), then there is a number c in (a, b) such that f 0 (c) = (f (b) f (a))/(b a).
FTC1: Suppose f is continuous on [a, b] and define G(x) =
G0 (x) = f (x). In other words, G is an antiderivative of f .
FTC2: Suppose f is continuous on [a, b]. Then
where F is any antiderivative of f .
Z
Z
x
f (t) dt. Then
a
b
f (x) dx = F (b)
a
F (a),