Differential Calculus
Definition
ƒ
'
(
x
) =
and
if this limit exists
ƒ
'
(
c
) =
If
ƒ
is differentiable at
x
=
c
, then
ƒ
is continuous at
x
=
c
.
Differentiation Rules
General and Logarithmic Differentiation Rules
1.
[
cu
] =
cu'
2.
[
u v
] =
u' v'
sum rule
3.
[
uv
] =
uv'
+
vu'
product rule
4.
[
] =
quotient rule
5.
[
c
] = 0
6.
[
un
] =
nun
1
u'
power rule
7.
[
x
] = 1
8.
[ln
u
] =
9.
[
eu
] =
euu'
10.
[
ƒ
(g(
x
))] =
ƒ
'
(
g
(
x
))
g'
(
x
)
chain rule
Derivatives of the Trigonometric Functions
1.
[sin
u
] = (cos
u
)
u'
2.
[csc
u
] = (csc
u
cot
u
)
u'
3.
[cos
u
] = (sin
u
)
u'
4.
[sec
u
] = (sec
u
tan
u
)
u'
5.
[tan
u
] = (sec2
u
)
u'
6.
[cot
u
] = (csc2
u
)
u'
Derivatives of the Inverse Trigonometric Functions
1.
[arcsin
u
] =
2.
[arccsc
u
] =
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
3.
[arccos
u
] =
4.
[arcsec
u
] =
5.
[arctan
u
] =
6.
[arccot
u
] =
Implicit Differentiation
Implicit differentiation
is useful in cases in which you cannot easily solve for y as a function of
x
.
Exercise
:
Find
for
y
3 +
xy
 2
y

x
2 = 2
[
y
3 +
xy
 2
y

x
2] =
[2]
3
y
2
+ (
x
+ y
)  2
 2
x
= 0
(3
y
2 +
x
 2) = 2
x

y
=
Higher Order Derivatives
These are successive derivatives of
ƒ
(
x
). Using prime notation, the second derivative of
ƒ
(
x
),
ƒ
''
(
x
), is the derivative
of
ƒ
'
(
x
). The numerical notation for higher order derivatives is represented by:
ƒ
(
n
)(
x
) =
y
(
n
)
The second derivative is also indicated by
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 EDELSTEIN
 Calculus, Derivative, Differential Calculus, lim, higher order derivatives, rule Quotient rule

Click to edit the document details