From the equation
, we can find
in terms of
. In other words, we can find
. In general, given a quadratic function
, if
always satisfies
, then we say the equation
is an implicit function determined by the equation
.
Understanding |
---|
Finding the implicit function determined by the equation
is the same as solving the equation
for . But for some type of and some , there may not be a satisfying the equation
.
|
Theorem 4..8 Suppose that

is the class

at

in the region

. If
, then there is an unique implicit function

determined by

in the neighborhood of

which satisfies
is a funciton of
and
is a function of
. Thus we have the following figure.
Thus,
NOTE In the neighborhood of the point satisfying
, the implicit function
exists and the implcit function is differentiable. Thus the total diffetential of
is
Now using the condition
,
This is 3. Furthermore, differentiate the above equation with respect to
, we have
Now we use
to obtain
Implicit Functions |
---|
is called 2nd order derivative of implicit function
|
Example 4..19 Find the derivative of a implicit function

such as

deteremined by
SOLUTION
Set
and take total differential of
,
Thus,
Since
,
A problem of finding a differentiation of an implicit function is solved by taking total differentiatial
Check |
---|
|
Exercise 4..19 Find the derivative of a implicit function

such as

deteremined by
Exercise4-19 |
---|
Since
and
.
|
SOLUTION Set
and find total differential of
. Then
Thus,
Example 4..20 Find

for the implicit function

determined by the equation

.
By writing
, we can find
.
SOLUTION
For
, set
. Then,
Thus,
Similarly,
Exercise 4..20 Find

for the implicit function

determined by the equation

.
SOLUTION
For
, set
.
Thus,
Similarly,
Theorem 4..9 If an implicit function

of

is determined by the equations

, then,
Theorem4-9 |
---|
To find
, we first find and then eliminate .
|
Example 4..21 Find

for the implicit functions

of

determined by the equations

.
SOLUTION
Let
. Then totally differentiate
and
to get
Now delete
to get
Check |
---|
From the equations
, we delete by adding
to the first equation,
.
|
Thus,
Similarly, delete
to get
Thus,
Exercise 4..21 Find

for the implicit functions

of

determined by the equations

.
Check |
---|
. Eliminate by multiplying to the former equation and multiplyimg to the latter equation.
. Simplifying,
|
SOLUTION Let
. Then find total derivatives.
Now using the equation 4.3 and the equation 4.4 to eliminate
.
Then
This time, using the equation 4.3 and the equation 4.4 to eliminate
.
Thus
- 1.
- Find the
for the implicit functions
of
determined by the following functions.
(a)
(b)
(c)
(d)
- 2.
- Find the
for the implicit functions
of
determined by the following functions.
(a)
(b)
- 1.
- Find
for the implicit function
of
determined by the following equations.
(a)
(b)
(c)
(d)
- 2.
- Find
for the implicit function
of
determined by the following equations.
(a)
(b)
- 3.
- Find the equation of the tangent line and the normal line to the curve
at the point
.
- 4.
- Find the equation of the tangent plane to the surface
at the point
.Find the equation of the normal line through
.
- 5.
- Find the local extrema of
implicitly defined by the following equations.
(a)
(b)
(c)