Class
Meetng#\Z
Thp Chaln
Rule
-The
chain rule
involves
multiplying
and/or
dividing
certain
derivatives
in
order
to obtain a desired derivative.
-This
rule
will
enable
us
to
expand the formulas
for
derivatives.
^Composite
functions:
In simple terms,
a
composite
function is form by
inserting
a function
into
the
domain
of
another function.
The
one
inserted is
the
"inside"
function & the other
is
the
"outside"
function. See
my
discussion
on
my website.
-So,
if
y
=
f(u)
and
u
=
g(x),
then
y
is
the composite
f[g(x)].
-Example:
Let
(x)
=
*'* 2x
+l
and
g(x)
=
3x
-
7. Form
the composites
f[g(x)] and
gffix)].
The
eeneral
power,
ryle
for
differentiation:
form:
y
=
rr'
,
where u
is
a function of
x.
Let
y-,r',
u=g(x). Want
dy1dx.
Sincey.n
"-+dy/du:nu'-'.
Sinceu:g(x) -+dr.r/dx=g'(x).So,
dy/dx:(dy/du[du/dx)=n u'-'du/dt
Example: Do f(x)
:
1x'
+2x+1)'
Example:
The
derivative ofy:
e'(covered on class
meeting #11).
Review it.
The
deri,vative
of v
=
[n
u:
Let
y*lnq
where u=g(x). Want dyldx
Since
5lnu
+ dyidu
*
1/u. Since
u:g(x)
-+
du/dx:
g
'(x),
so, dy/dx
=(dyldu[du/dx)
:
(l/uXdt/dx)
Example;
Do
f(x):ln(
}-x-x'
)
Examples ofusing
more
than one rule
po p:1l+lnx)o'5
Do
(x):
ln(l-e
-')
Example
problFm:
If
f(tF1,0o0e''o*'
Represents
the balance at
time t
when a
deposit
of
$1,000
is
compounded
continuously,
find
how fast
the balance is
growing
after
10
yems?
Want f
'(10)
