Calculus Cheat Sheet
Visit
http://tutorial.math.lamar.edu
for a complete set of Calculus notes.
©
2005 Paul Dawkins
Derivatives
Definition and Notation
If
()
y fx
=
then the derivative is defined to be
(
)
0
lim
h
fxh
f
x
fx
h
→
+−
′
=
.
If
yfx
=
then all of the following are
equivalent notations for the derivative.
df
dy
d
f
xy
f
x D
f
x
dx
dx
dx
′′
== = =
=
If
=
all of the following are equivalent
notations for derivative evaluated at
x
a
=
.
xa
df
dy
f
ay
D
f
a
dx
dx
=
==
=
=
Interpretation of the Derivative
If
yf
x
=
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
(
)
y fa f a x a
′
=+
−
.
2.
(
)
f
a
′
is the instantaneous rate of
change of
(
)
f
x
at
x
a
=
.
3.
If
(
)
is the position of an object at
time
x
then
(
)
f
a
′
is the velocity of
the object at
x
a
=
.
Basic Properties and Formulas
If
f
x
and
g
x
are differentiable functions (the derivative exists),
c
and
n
are any real numbers,
1.
(
)
c f
c f
x
′
′
=
2.
(
)
fg f
x
g
x
′
±=
±
3.
f
gf
g
f
g
′
– Product Rule
4.
2
f
fg fg
gg
′
⎛⎞
−
=
⎜⎟
⎝⎠
– Quotient Rule
5.
0
d
c
dx
=
6.
1
nn
d
x
nx
dx
−
=
– Power Rule
7.
d
f gx
f gx g x
dx
=
This is the
Chain Rule
Common Derivatives
1
d
x
dx
=
sin
cos
d
x
x
dx
=
cos
sin
d
x
x
dx
=−
2
tan
sec
d
x
x
dx
=
sec
sec tan
d
x
xx
dx
=
csc
csc cot
d
x
dx
2
cot
csc
d
x
x
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
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
d
nfx
dx
−
=
⎡⎤
⎣⎦
2.
(
)
f
xf
x
d
dx
′
=
3.
ln
f
x
d
dx
f
x
′
=
4.
sin
cos
d
f
x
f
x
f
x
dx
′
=
5.
cos
sin
d
f
x
f
x
f
x
dx
′
6.
2
tan
sec
d
f
x
f
x
dx
′
=
7.
[]
[][]
sec
sec
tan
f
x f
x
f
x
d
dx
′
=
8.
(
)
1
2
tan
1
d
dx
f
x
−
′
=
+⎡
⎤
Higher Order Derivatives
The Second Derivative is denoted as
2
2
2
df
fx f x
dx
and is defined as
f
x
′
′
=
,
i.e.
the derivative of the
first derivative,
f
x
′
.
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 Spring '08
 Skrypka
 Calculus, Derivative, Paul Dawkins, dx dx dx, Calculus Cheat Sheet

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