Implicit
Differentiation
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Objectives:
Students
will
be
able
to
• Calculate
derivative
of
function
defined
implicitly.
• Determine
the
slope
of
the
tangent
line
to
a
function
defined
implicitly
at
a
specified
point.
• Calculate
the
equation
of
the
tangent
line
to
a
function
defined
implicitly
at
a
specified
point.
All
of
our
functions
so
far
have
explicitly
define
y
as
a
function
of
x.
Not
all
functions
can
be
easily
defined
in
this
way.
Instead
we
can
have
functions
that
implicitly
define
y
as
a
function
of
x.
Such
implicitly
defined
functions
can
also
be
differentiated.
The
process
will
involve
the
use
of
the
chain
rule.
The
thing
to
keep
in
mind
is
that
y
is
a
function
of
x
and
d ( f ( x))
thus
can
be
written
as f ( x) .
The
derivative
of
f ( x) is
.
Since
f ( x) is
y,
dx
d ( y)
this
can
be
rewritten
as
or
y ′.
This
will
be
important
in
our
process
of
implicit
dx
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differentiation.
€
Example
1:
€
€
dy
2
2
Find
for
6x + 5y = 36 .
dx
Example
2:
€ dy
Find
for
7x 2 = 5y 2 + 4xy + 1.
dx
Example
3:
€ dy for
e x 3 y = 5x + 4 y + 2 .
Find
dx
Example
4:
3
5
€ dy for
Find
y ln( x) + 2 = x 2 y 2 .
dx
Example
5:
3
2
€
Find
the
equation
of
the
tangent
line
to
the
curve
2xe xy = e x + ye x at
the
point
(1,
1)
Example
6:
Find
the
equation
of
the
tangent
line
to
the
curve
y 3 + xy 2 + 1 = x + 2 y 2 at
x = 2 €
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