mat 1234 calculus i section 2.6 implicit differentiation
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MAT 1234Calculus I
Section 2.6
Implicit Differentiation
http://myhome.spu.edu/lauw
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Introduce a new form of extended power rule (notations changes)
Introduce Implicit Differentiation
Recall (Extended Power Rule)
dx
dunu
dx
dy
xguuy
xgy
n
n
n
1
)( ,
)(
dx
dunuu
dx
d nn 1
Recall (Extended Power Rule)
dx
dunu
dx
dy
xguuy
xgy
n
n
n
1
)( ,
)(
dx
dunuu
dx
d nn 1
Recall (Extended Power Rule)
dx
dunu
dx
dy
xguuy
xgy
n
n
n
1
)( ,
)(
dx
dunuu
dx
d nn 1
Recall (Extended Power Rule)
dx
dunuu
dx
d nn 1
We now free the variable , which we need for the next formula.
Recall (Extended Power Rule)
If is a function in , then
dx
dunuu
dx
d nn 1
If y is a function in x, then
dx
dynyy
dx
d nn 1
Recall (Extended Power Rule)
If is a function in , then
dx
dunuu
dx
d nn 1
If is a function in , then
dx
dynyy
dx
d nn 1 Wai, do what you need to do.
Example 0
5dy
dx
dx
dynyy
dx
d nn 1
The Needs for Implicit Differentiation…
Example 1
Find the slopes of the tangent line on the graph
i.e. find
122 yx
x
y
),( yxdx
dy
Example 1: Method I
Make y as the subject of the equation:
122 yx
x
y
2
22
1
1
xy
xy
21 xy
Example 1: Method I
Make y as the subject of the equation:
122 yx
x
y
2
22
1
1
xy
xy
21 xy
Example 1: Method I
Make y as the subject of the equation:
122 yx
x
y
2
22
1
1
xy
xy
21 xy
Example 1: Method I
Suppose the point is on the upper half circle
122 yx
x
y
2
2
22
12
2
122
1
)2(12
1
)1()1(2
1
)1(1
x
x
xx
xdx
dx
dx
dy
xxy
),( yx
Example 1: Method I
Suppose the point is on the lower half circle
122 yx
x
y
2
2
22
12
2
122
1
)2(12
1
)1()1(2
1
)1(1
x
x
xx
xdx
dx
dx
dy
xxy
),( yx
Example 1: Method I
Two disadvantages of Method I:
1. ???
2. ???
122 yx
Example 1: Method II
Implicit Differentiation:
Differentiate both sides of the equation.
122 yx
dx
dynyy
dx
d nn 1
2 2 1d dx y
dx dx
Expectation
You are required to show the implicit differentiation step
2 2 1d dx y
dx dx
Notations
(…) is the derivative of (…). Do not confuse it with which is the derivative of .
2 2 1x yd d
dx dx
Example 2 xyyx 233
3 3 2d dx y xy
dx dx
dx
dynyy
dx
d nn 1
Example 3
Find the slope of the tangent line at . 1,1
xyyx 233
Notations
Correct or
Incorrect
because
2
2, 1,1
2 1 3 1
3 1 2 1x y
dy
dx
2
2
2 1 3 1
3 1 2 1
dy
dx
2
21,1
2 1 3 1
3 1 2 1
dy
dx
2
2
2 3
3 2
dy y x
dx y x
Example 4xyx )cos(
cos( )d d
x y xdx dx
dx
dynyy
dx
d nn 1