Stuff you MUST know cold for AP Calculus!
Note: Letters like a, b, c, d, m, and n are traditionally used to represent constants.
Letters like f, g, h, u, v, x, and y and traditionally used to represent variables or functions.
Basic Derivatives
 
Definitions of Derivative
f x  h   f x 
h 0
h
f  x   f c 
lim
x c
xc
d n
x  nx n 1
dx
d
sin x  cos x
dx
d
cos x   sin x
dx
d
tan x  sec2 x
dx
d
cot x   csc2 x
dx
d
sec x   sec x tan x
dx
d
csc x   csc x cot x
dx
d
ln x  1
dx
x
d x
e   e x
dx
lim
Differentiation Rules
Chain Rule
If f x   g hx  , then
f x   g hx   hx  .
OR
dy dy du


dx du dx
Product Rule
d
uv  v du  u dv
dx
dx
dx
Quotient Rule
d  u  vu  uv
 
dx  v 
v2
More Derivatives
d
sin 1 x  1 2
dx
1 x
d
1
cos x  1 2
dx
1 x
d
tan1 x  1 2
dx
1 x
d
1
cot x  1 2
dx
1 x
d
sec1 x  12
dx
x x 1
d
csc1 x  12
dx
x x 1
d x
a   a x ln a
dx
d
log a x   1
dx
x ln a
If a function is differentiable,
then it is continuous.
“PLUS A CONSTANT!”
The Fundamental Theorem of
Calculus
b
 f x dx  F b   F a 
a
where F x   f x 
Rolle’s Theorem
If f is differentiable for all values of x
in the open interval a, b  , and f is
continuous at x  a and at x  b , and
f a   f b   0 ,
then there is at least one number x  c
in a, b  such that f c   0 .
Mean Value Theorem
If f is differentiable for all values of x
in the open interval a, b  , and f is
continuous at x  a and x  b , then
there is at least one number x  c in
a, b  such that f c   f b  f a 
ba
Trapezoidal Rule
1ba
 f x 0 
n 
b
 f x dx  2 
a
 2 f x1     2 f x n 1   f x n 
Distance, Velocity, and Acceleration
If distance, velocity and acceleration
are represented by s, v, and a,
dv d 2 s
respectively, then a 
and

dt dt 2
ds
v .
dt
t1
change in v   a dt
t0
Corollary to the FTC
d
dx
g x

f t dt  f g x   g x 
a
t1
change in s   v dt
t0
Intermediate Value Theorem
If function f is continuous for all x in
the closed interval a, b , and y is a
number between f a  and f b , then
there is a number x  c in a, b  for
which f c   y .
l'Hôpital’s Rule
If lim
x a
then lim
xa
f x  0 
 or
g x 
0 
f x 
f x 
 lim
xa
g x 
g x 