4.3 - MATH
1081
 Wednesday,
February
16
 


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Unformatted text preview: MATH
1081
 Wednesday,
February
16
 
 Chapter
4
–
Section
3
 
 THE
CHAIN
RULE
 
 Homework
#5
(due
2/21):
 Section
4.2
#8,
22,
32,
46,
52
 
 
 
 
 
 
 
 
 Section
4.3
#34,
38,
56,
62
 
 Clicker
Check­in:

Choose
any
letter
to
check
in
now.
 
 Example:

Suppose
that
 f ( x ) 
and
 g ( x ) 
are
differentiable

 
functions
at
 x = 2 
with

 f ( 2 ) = 7 ,
 f ' ( 2 ) = 3,
 g ( 2 ) = 5 ,
and
 g ' ( 2 ) = −4 .


 
 5 f ( x) 1. Find
 q ' ( 2 )
when
 q ( x ) = .
 g ( x ) − 10 
 
 
 
 2. Find
 p ' ( 2 ) 
when
 p ( x ) = x 2 f ( x ) .
 
 
 Consider
the
function
 f ( x ) = x − 5 .

At
this
point,
to
find
the
 derivative
of
this
function,
we
would
either
need
to
multiply
it
 out
to
get
the
polynomial
 f ( x ) = x 6 − 10 x 3 + 25 
or
write
it
as
a
 product
 f ( x ) = x 3 − 5 x 3 − 5 .
 
 For
this
particular
function,
either
of
these
options
is
fairly
 simple.

However,
as
soon
as
we
either
start
to
increase
the
 21 3 power
on
the
outside
to
 f ( x ) = x − 5 
or
we
make
the
 function
on
the
inside
more
complicated
like
 2 3 2 ⎛ x + 4x − 5⎞ ,
doing
either
of
the
above
options
 f ( x) = ⎜ 1/ 2 ⎟ ⎝ 9x − 8x ⎠ becomes
more
and
more
complicated.
 3 2 ( ) ( )( ) ( ) Today,
we
will
look
at
the
chain
rule
that
will
help
us
with
 these
more
complicated
functions.

In
particular,
the
chain
rule
 shows
us
how
to
find
the
derivative
of
a
composite
function.
 
 Definition:

Composite
function
 
 Let
 f 
and
 g 
be
functions.

The
composite
function,
or
 composition,
of
 g 
and
 f 
is
the
function
whose
values
are
 given
by
 g ⎡ f ( x ) ⎤ 
for
all
 x 
in
the
domain
of
 f 
for
which
the
 ⎣ ⎦ value
 f ( x ) 
is
in
the
domain
of
 g .
 
 In
other
words,
a
composite
function
is
one
built
by
plugging
 one
function
into
another.

Plug
an
 x 
into
 f 
and
take
the
 y 
that
 comes
out
of
 f 
and
plug
that
into
 g .

The
value
that
comes
out
 of
 g 
is
the
result.
 Example:

Find
the
compositions
 g ⎡ f ( x ) ⎤ 
and
 f ⎡ g ( x ) ⎤ 
of
the

 ⎣ ⎦ ⎣ ⎦ 
functions.
 
 x 
 
 
 1.

 f ( x ) = −8 x + 9 
and
 g ( x ) = + 4 
 5 

 
 
 
 
 
 2.

 f ( x ) = 9 x 2 − 11x 
and
 g ( x ) = 2 x + 2 
 
 
 
 
 
 
 3.

 f ( x ) = x 21

and

 g ( x ) = x 3 + 4 x 2 − 5 
 Rules
of
Differentiation
 
 
 7.

Chain
Rule
 
 
 
 If
 f ( x ) = u ⎡ v ( x ) ⎤ ,
then
 f ' ( x ) = u ' ⎡ v ( x ) ⎤ ⋅ v ' ( x ) ,
assuming

 ⎣ ⎦ ⎣ ⎦ 
 
 all
relevant
derivative
exist.
 
 
 In
other
words,
take
the
derivative
of
the
outside
function,
 leaving
the
inside
alone.

Then,
take
the
derivative
of
the
inside
 function.

So,
we
are
creating
a
“chain”
working
from
the
 outside
in.
 
 Example:

Find
the
derivative
of
the
function.
 
 2 3 
 
 
 1.

 f ( x ) = x − 5 
 
 
 
 3/ 2 2 
 
 
 2.

 g ( t ) = 5t + 9t − 8 
 
 
 
 
 x +1 
 
 
 3.

 y = 5 

 3x 2 − 11x + 4 ( ) ( ) ( ) Example:

Consider
the
table
of
values
of
functions
and
their

 
derivatives
at
various
points.
 
 x
 1
 2
 3
 4
 f ( x)
 f '( x )
 g ( x)
 g '( x )
 2
 ‐6
 2
 2/7
 4
 ‐7
 3
 3/7
 1
 ‐8
 4
 4/7
 3
 ‐9
 1
 5/7
 
 Find
 Dx f ⎡ g ( x ) ⎤ and
 Dx g ⎡ f ( x ) ⎤ 
at
 x = 1
and
 x = 2 .
 ⎣ ⎦ ⎣ ⎦ ( ) ( ) Clicker
Check­out:

Choose
any
letter
to
check
out
now.
 
 
 Tomorrow
in
recitation:

Workshop
#3
 
 Next
time
–
Section
4.4

 
 ...
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