MAT 1234 Calculus I Section 2.5 Part II Chain Rule .
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Transcript of MAT 1234 Calculus I Section 2.5 Part II Chain Rule .
Part II
WebAssign HW 2.5 Part II (Do it early) Extended Power Rule –a special case of
the chain rule Higher Derivatives (2.2)
Extended Power Ruleu
Extended Power Rule
11
( )
, ( )
( ) ( )
n
n
nn
y g x
y u u g x
dy du dynu OR n g x g x
dx dx dx
Example 8
5 32x x
d
dx
4 25 6x x
320 12x x
d
dx
5 3First derivative of 2x x
5 3Second derivative of 2x x
Given a function f(x)
which is a function.
)( of derivativefirst the
)( of derivative the)(
xf
xfxf
Higher Derivatives
Given a function f’(x)
which is a function.
)( of derivative second the
)( of derivative the
)()(
xf
xf
xfdx
dxf
Higher Derivatives
Given a function f’’(x)
which is a function.
)( of derivative third the
)( of derivative the
)()(
xf
xf
xfdx
dxf
Higher Derivatives
In general, we denote the n-th derivative of f(x) as
)]([ , , ),( )()( xfDdx
ydyxf n
n
nnn
Higher Derivatives
Notations
,,,,,
),(),(),(),(),(
,,,,,
)(
5
5
4
4
3
3
2
2
)5()4(
)5()4(
dx
yd
dx
yd
dx
yd
dx
yd
dx
dy
xfxfxfxfxf
yyyyy
xfy
,,,,,
),(),(),(),(),(
,,,,,
)(
5
5
4
4
3
3
2
2
)5()4(
)5()4(
dx
yd
dx
yd
dx
yd
dx
yd
dx
dy
xfxfxfxfxf
yyyyy
xfy
Notations
,,,,,
),(),(),(),(),(
,,,,,
)(
5
5
4
4
3
3
2
2
)5()4(
)5()4(
dx
yd
dx
yd
dx
yd
dx
yd
dx
dy
xfxfxfxfxf
yyyyy
xfy
Notations