Lecture 6 : Friday April 10th
[email protected]
Key concepts : Vector valued function, gradient, differentiability,
properties, chain rule
6.1 Basic Properties of derivatives.
For functions of one variable
f
:
R
→
R
, the basic properties of derivatives stem from the
basic properties of limits, namely the sum, product and quotient rules, as well as the chain
rule. The difference for functions of more than one variable is that for these properties to
hold, the function is required to be
differentiable
. If the function is not differentiable then
the rules can fail. Here are the basic properties of derivatives. For all of the following
rules,
f, g
:
R
n
→
R
m
are defined on an open set
U
and differentiable functions at a point
a
∈
U
. For the quotient rule,
g
is nonzero on
U
.
•
Constant multiple rule:
cf
(
x
) is differentiable at
a
and
∇
(
cf
)(
a
) =
c
∇
f
(
a
)
.
•
Sum rule:
f
(
x
) +
g
(
x
) is differentiable at
a
and
∇
(
f
+
g
)(
a
) =
∇
f
(
a
) +
∇
g
(
a
)
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 Enright
 Math, Chain Rule, Derivative, Rn Rm

Click to edit the document details