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MAT 1234Calculus I
Section 2.5 Part II
Chain Rule
http://myhome.spu.edu/lauw
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 Rule
dx
dunu
dx
dy
xguuy
xgy
n
n
n
1
)( ,
)(
u
Extended Power Rule
dx
dunu
dx
dy
xguuy
xgy
n
n
n
1
)( ,
)(
u
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
2 3( 1)y x x
Example 4
1( ) ( )
ndyn g x g x
dx
Example 5
xxy 44 cossin
1( ) ( )
ndyn g x g x
dx
More Examples on Chain Rules
Example 6 (Product Rule, Chain rule, & Power
Rule)
11sin siny x x x
x
Example 7 (Quotient Rule, & Ext. Power Rule) 3
1
xy
x
1( ) ( )
ndyn g x g x
dx
Higher Derivatives
Example 8
5 32x x
d
dx
4 25 6x x
320 12x x
d
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
Example 9
Find if xxy siny
Example 10
