This preview shows pages 1–3. Sign up to view the full content.
Calculus Cheat Sheet
Visit
http://tutorial.math.lamar.edu
for a complete set of Calculus notes.
© 2005 Paul Dawkins
Derivatives
Definition and Notation
If
()
yf
x
=
then the derivative is defined to be
( ) ( )
0
lim
h
fx h fx
fx
h
→
+−
′
=
.
If
yfx
=
then all of the following are
equivalent notations for the derivative.
df
dy
d
fx y
D
dx
dx
dx
′′
==
=
=
=
If
( )
=
all of the following are equivalent
notations for derivative evaluated at
x
a
=
.
xa
df
dy
fa y
D
fa
dx
dx
=
=
=
Interpretation of the Derivative
If
=
then,
1.
mfa
′
=
is the slope of the tangent
line to
=
at
x
a
=
and the
equation of the tangent line at
x
a
=
is
given by
() ()
( )
yfa faxa
′
=+
−
.
2.
( )
′
is the instantaneous rate of
change of
( )
at
x
a
=
.
3.
If
( )
is the position of an object at
time
x
then
′
is the velocity of
the object at
x
a
=
.
Basic Properties and Formulas
If
and
gx
are differentiable functions (the derivative exists),
c
and
n
are any real numbers,
1.
(
)
cf
cf x
′
′
=
2.
( ) () ()
fg fx gx
′
±=
±
3.
fg
f g fg
′
– Product Rule
4.
2
ff
g
f
g
gg
′
⎛⎞
−
=
⎜⎟
⎝⎠
– Quotient Rule
5.
0
d
c
dx
=
6.
1
nn
d
xn
x
dx
−
=
– Power Rule
7.
d
fgx
f gx gx
dx
=
This is the
Chain Rule
Common Derivatives
1
d
x
dx
=
sin
cos
d
xx
dx
=
cos
sin
d
dx
=−
2
tan
sec
d
dx
=
sec
sec tan
d
x
dx
=
csc
csc cot
d
x
dx
2
cot
csc
d
dx
1
2
1
sin
1
d
x
dx
x
−
=
−
1
2
1
cos
1
d
x
dx
x
−
−
1
2
1
tan
1
d
x
dx
x
−
=
+
ln
d
aaa
dx
=
d
dx
=
ee
1
ln
,
0
d
dx
x
=>
1
ln
,
0
d
dx
x
=≠
1
log
,
0
ln
a
d
dx
x
a
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Calculus Cheat Sheet
Visit
http://tutorial.math.lamar.edu
for a complete set of Calculus notes.
© 2005 Paul Dawkins
Chain Rule Variants
The chain rule applied to some specific functions.
1.
()
( )
() ()
1
nn
d
fx
nfx
f x
dx
−
′
=
⎡⎤
⎣⎦
2.
d
dx
′
=
ee
3.
ln
d
dx
f x
′
=
4.
sin
cos
d
dx
′
=
5.
cos
sin
d
dx
′
=−
6.
2
tan
sec
d
dx
′
=
7.
[]
sec
sec
tan
d
dx
′
=
8.
1
2
tan
1
d
dx
−
′
=
+⎡
⎤
Higher Order Derivatives
The Second Derivative is denoted as
2
2
2
df
fx f x
dx
′′
==
and is defined as
′
′
=
,
i.e.
the derivative of the
first derivative,
′
.
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 01/17/2011 for the course CE CE taught by Professor Armstrong during the Spring '10 term at The University of Texas at San Antonio San Antonio.
 Spring '10
 ARMSTRONG

Click to edit the document details