Dif

บทท่ี2 เร่ืองอนุพันธของฟงกชัน

Calculus 1Calculus 1Calculus 1Calculus 1

1

ก�� )(xfy = �� ก�� ),( yxP �� )(xfy = �� ),( hyhxQ ++ �� !"��ก�ก�� P ��$!�%"�&��'(

)'�� x ��&� )(xfy = )'�� hx + ��&� )( hxfy +=

��ก��

1. ��%$�ก�$��'� �� 2. ��-��.�/�� ก�� 3. ก�$��-��.� ��ก��-'��12%�� 2 �3 4. ก�$��-��.� ��ก��-'��12%�� !%$ 5. ก�$��-��.� ��ก��26� �� !%$ 6. ��-��.� ��ก��$�ก�� ก�� $2 � ��-��.��!�

�� ก �� 2

��ก��

),( yxP ∆x

y∆∆

),( hyhxQ ++

hx + x x

y

O

��ก�� :� ��ก��)'�/' �&�� 2-�� ;��$�ก��< �"��%�< %�<��$��$:��3� �<ก �� $ ก$1=� ��' �"�

155 +− xx ��%�



2

��(��"�)'��'� ��ก�� x &� �� hx + $' ก<"� ��%$�ก�$��'� �� /��%�<��$ x ��&�

xhxx −+=∆ )( ��)��&� )(xf ��'� �&�� )( hxf + $' ก<"� ��%$�ก�$��'� �� /�� ก��&�

)()( xfhxfy −+=∆ ��%$�ก�$��'� ��/�� y )' �ก�� x �:� ��%$��"<�/��"� y )'��'� �&� %"��"� x )'��'� �

h

xfhxf

x

y

xhx

xfhxf

x

y

)()(

)(

)()(

−+=

∆∆

−+−+

=∆∆

3:�� x∆ /��ก� 0 ��"�&� $��$' ก�"��232%�'(<"� �� ก�� )(xfy = ��ก�� x

�:� h

xfhxfh

)()(lim

0

−+→

�$:� x

yx ∆∆

→∆ 0lim

�� 73 2 −= xy ��%$�ก�$��'� ��/�� y )' �ก�� x �� ?�'� ��"<�3:�� x ��'� ��ก 1 &�� 4 ��!� ��ก��) ��&� 4=+ hx , 1=x ��%$�ก�$��'� ��

14

)1()4(

)(

)()(

−−

=

−+−+

=∆∆

ff

xhx

xfhxf

x

y

3

)4(413

)7)1(3()7)4(3( 22

−−=

−−−=

153

45

=

=

��(��%$�ก�$��'� ��/�� y )' �ก�� x �� ?�'� ��"<�3:�� x ��'� ��ก 1 &�� 4 )"�ก�� 15

��%$�ก�$��'� ��/�� y )' �ก�� x �� ?�'� ��"<�3:�� x ��'� ��ก 1 &�� 4 )"�ก�� 15 �� "# � x�� "�$%� 1 '� ��(� y )*

�� "�$%��*"�� 15 '� ��



3

��ก��

��"�� ก�� )(xfy = �� ก�� -��.�/�� )(xf /' ��)��< ��A��กB1� )(xf ′ �"��<"� Cf prime xJ ;��ก��/�(��'(

h

xfhxf

x

yxf

hx

)()(limlim)(

00

−+=

∆∆

=′→→∆

K��232%�'(��"�&�$��ก�"�<<"� f �� ก��)'��-��.�&� ��$' ก )(xf ′ <"� C��-��.�/�� ก�� f )'� x J K� )(xfy = ��< ��-��.�/�� ก�� f )'� x ��ก��ก��/' ��)��< )(xf ′ ��3�$K/' ��)��A��กB1��:��&��'ก �:�

dx

dyy ,′ �$:�

dx

xdf )(

��ก�� f �� x ��ก��ก�� )(xfy = � �� ! "�#��ก��



4

�� ก�� 3)( += xxf ��-��.�/�� )(xf

��!� ��ก��) � 3)( += xxf ��&� 3)()( ++=+ hxhxf ��ก�2 �3

1

lim

33lim

)3()3)((lim

)()(lim)(

0

0

0

0

=

=

−−++=

+−++=

−+=′

→

→

→

→

h

hh

xhxh

xhxh

xfhxfxf

h

h

h

h

��(��-��.�/�� 3)( += xxf )"�ก�� 1

ก��'��ก��,-�.�(��"

��</��'($��2 �3/��-��.�-:��<1��"��-��.��:�

h

xfhxfxf

h

)()(lim)(

0

−+=′

→



5

�� ก�� 1)( 2 −+= xxxf ��-��.�/�� )(xf

��!� ��ก��) � 1)( 2 −+= xxxf ��&� 1)()()( 2 −+++=+ hxhxhxf

1)()2( 22 −++++= hxhxhx (�!%$ก��3�!$1� ( ) 222 2 BABABA ++=+ )

)1()12(

122

22

−++++=

−++++=

xxhxh

hxhxhx

��ก�2 �3

[ ]

12

102

12lim

)12(lim

)1()1()12(lim

)()(lim)(

0

0

22

0

0

+=

++=

++=

++=

−+−−++++=

−+=′

→

→

→

→

x

x

hxh

hxhh

xxxxhxh

h

xfhxfxf

h

h

h

h

��(��-��.�/�� 1)( 2 −+= xxxf )"�ก�� 12 +x

)�)<��!%$�!%$ก��3�!$1� ( ) 222 2 BABABA ++=+ ( ) 222 2 BABABA +−=−



6

�� ก�� 2)( 3 −= xxf ��-��.�/�� )(xf

��!� ��ก��) � 2)( 3 −= xxf ��&� 2)()( 3 −+=+ hxhxf

2)2)(( 22 −+++= hxhxhx ( )[ ]23 )BA)(BA(BA ++=+

)2()33(

233322

3223

−+++=

−+++=

xhxhxh

hxhhxx

��ก�2 �3

[ ]

2

2

22

0

22

0

3322

0

0

3

003

33lim

)33(lim

)2()2()33(lim

)()(lim)(

x

x

hxhx

h

hxhxh

h

xxhxhxh

h

xfhxfxf

h

h

h

h

=

++=

++=

++=

−−−+++=

−+=′

→

→

→

→

��(��-��.�/�� 2)( 3 −= xxf )"�ก�� 23x

3323

223

33

)2)(()(

BABBAA

BABABABA

+++=

+++=+



7

�� ก�� 0;1

)( >= xx

xf ��-��.�/�� )(xf

��!� ��ก��) � x

xf1

)( =

��&� hx

hxf+

=+1

)(

��ก�2 �3

xx

xxxx

xxxx

hxxhxx

hxxhxx

h

h

hxxhxx

hxx

h

hxx

hxx

hxx

hxx

h

hxx

hxx

h

xhxh

h

xfhxfxf

h

h

h

h

h

h

h

2

1

1

)0()0(

1

)()(

1lim

)()(

1lim

)()(

)(1lim

))((

1lim

)()(

1lim

111lim

)()(lim)(

0

0

0

0

0

0

0

−=

+

−=

+++

−=

+++

−=

+++

−=

+++

+−=

++

++

+

+−=

+

+−=

−

+=

−+=′

→

→

→

→

→

→

→

��(��-��.�/�� 0;1

)( >= xx

xf )"�ก�� xx2

1−



8

�� -��.�/�� )(xf %"�&��'(�� 2 �3 1. xxf 3)( = 2. 2)( 3 −= xxf 3. 123)( 2 +−= xxxf PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP



9

PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP



10

ก��'��ก��,-�.�(/0��

ก�$��-��.�� 2 �3��(��ก�$ �"� �ก��)��' <��3�ก ��(��3'ก�$�$��!%$��$��-��.�/�� ก��-'��12%/�(� �)�ก�$��2 �3 �� )(,)( xvxu �� ก��/�� x

�� c �� "��-��.�/�� ก��/' ��)��< )(xf ′ , dx

dyy ,′

�$:� dx

xdf )(

�� cxf =)( ��&�

chxf =+ )(

h

xfhxfxf

h

)()(lim)(

0

−+=′

→

0

lim)(0

=

−=′

→ h

ccxf

h

��(��&��!%$

0=dx

dc

/0��-1ก2



11

�� xxf =)( ��&� hxhxf +=+ )(

h

xfhxfxf

h

)()(lim)(

0

−+=′

→

1

1lim

lim

lim)(

0

0

0

=

=

=

−+=′

→

→

→

h

h

h

h

hh

xhxxf

��(��&��!%$

1=dx

dx

��ก�� "�� 0��0 �� )()( xcfxg = 3:�� c ��"��%�< ��&� )()( hxcfhxg +=+

h

xghxgxg

h

)()(lim)(

0

−+=′

→

)(

)()(lim

)]()([lim

)()(lim)(

0

0

0

xfch

xfhxfc

h

xfhxfch

xcfhxcfxg

h

h

h

′=

−+=

−+=

−+=′

→

→

→

��(��&��!%$ 3:�� c ��"��%�<

dx

xdfc

dx

xdcf )()(=

/��"�ก



12

��ก�� nx ��* nu �� nxxf =)( 3:�� n ��/��<��$2�� 0n ≠ ��&� nhxhxf )()( +=+

h

xfhxfxf

h

)()(lim)(

0

−+=′

→

1

1

121

0

121

0

221

0

2221

0

0

0...0

...!2

)1(lim

]...!2

)1([

lim

...!2

)1(

lim

]...!2

)1([

lim

)(lim)(

−

−

−−−

→

−−−

→

−−

→

−−

→

→

=

+++=

++−

+=

++−

+=

++−

+=

−++−

++=

−+=′

n

n

nnn

h

nnn

h

nnn

h

nnnn

h

nn

h

nx

nx

hhxnn

nx

h

hhxnn

nxh

h

hhxnn

hnx

h

xhhxnn

hnxx

h

xhxxf

��(��&��!%$ ��

1−= nn

nxdx

dx

nxxf =)( 3:�� n ��/��<��$2�� 0n ≠ -$��K� 0=n ��)�� 1)( 0 == xxf

dx

dunu

dx

du nn

1−=



13

��3��ก'�#�3�� ก�� )()()( xvxuxf ±= �� )'� u �� v 3'��-��.�)'� x ��&� )()()( hxvhxuxf +±+=+

h

xfhxfxf

h

)()(lim)(

0

−+=′

→

)()(

)()(lim

)()(lim

)]()([)]()([lim)(

00

0

xvxuh

xvhxv

h

xuhxuh

xvxuhxvhxuxf

hh

h

′±′=

−+±

−+=

±−+±+=′

→→

→

��(��&��!%$

dx

dv

dx

duvu

dx

d±=± ][

�� -��.�/�� 7254)( 23 −+−= xxxxf

��!� )7254()( 23 −+−=′ xxxdx

dxf

7254 23

dx

dx

dx

dx

dx

dx

dx

d−+−= → ][ vu

dx

d±

7254 23

dx

dx

dx

dx

dx

dx

dx

d−+−= → )(xcf

dx

d

0)1(2)2(5)3(4 2 −+−= xx → dx

dc

dx

dx

dx

dxn

,,

21012 2 +−= xx ��(� ��-��.�/�� 7254)( 23 −+−= xxxxf )"�ก�� 21012 2 +− xx



14

�� -��.�/�� 96 )35()( −= xxf

��!� 96 )35()( −=′ xdx

dxf

)35()35(9 686 −−= xdx

dx → nu

dx

d

]35[)35(9 686

dx

dx

dx

dx −−= → ][ vu

dx

d± , )(xcf

dx

d

]0)6(5[)35(9 586 −−= xx → dx

dc

dx

dxn

,

865 )35(27 −= xx ��(� ��-��.�/�� 96 )35()( −= xxf )"�ก�� 865 )35(27 −xx

�� -��.�/�� 5

432)(

xxxxf +−=

��!� )43

2()(5xx

xdx

dxf +−=′

52

1

2

1

432 −−+−= x

dx

dx

dx

dx

dx

d → ][ vudx

d±

52

1

2

1

432 −−+−= x

dx

dx

dx

dx

dx

d → )(xcfdx

d

)5(4)2

1(3)

2

1(2 62

3

2

1−−−

−+−−= xxx → dx

dxn

62

3

2

1

202

3 −−−−+= xxx

6

2

3

20

2

31x

xx

−+=

��(� ��-��.�/�� 5

432)(

xxxxf +−= )"�ก��

6

2

3

20

2

31

xx

x−+



15

�� -��.�/��

1. 73 )43()( +−= xxxf 2. 522)( 3 ++= xxxf 3. 74

713)(

xxxxf −−=

4. 523 )943(

1)(

−−+=

xxxxf 5. 35335)( 81221 −+−−= xxxxxf

PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP



16

PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP



17

��3��0��ก�� )().()( xvxuxf = ��&� )().()( hxvhxuhxf ++=+

h

xvxuhxvhxuxf

h

)().()().(lim)(

0

−++=′

→

�� )x(v).hx(u + 3��<ก/��ก ��&�

h

xvhxuxvhxuxvxuhxvhxuxf

h

)]().()().([)]().()().([lim)(

0

+−++−++=′

→

h

xvxuxvhxuxvhxuhxvhxuh

)]().()().([)]().()().([lim

0

−+++−++=

→

h

xuhxuxvxvhxvhxuh

)]()()[()]()()[(lim

0

−++−++=

→

h

xuhxuxv

h

xvhxvhxu

hhhh

)]()([lim).(lim

)]()([lim.)(lim

0000

−++

−++=

→→→→

)().()(.)( xudx

dxvxv

dx

dxu +=

��(��&��!%$

dx

duv

dx

dvuvu

dx

d+=).(



18

��3�'��ก��

�� )(

)()(

xv

xuxf = ��&�

)(

)()(

hxv

hxuhxf

++

=+

)(.)(.

)(.)()(.)(lim

)(.)(

)(.)()(.)(

lim

)(

)(

)(

)(

lim)(

0

0

0

xvhxvh

xuhxvxvhxuh

xvhxv

xuhxvxvhxuh

xv

xu

hxv

hxu

xf

h

h

h

++−+

=

++−+

=

−++

=′

→

→

→

�� )().( xvxu 3��<ก/��ก ��&�

)(.)(.

)().()().()](.)()(.)([lim)(

0 xvhxvh

xvxuxvxuxuhxvxvhxuxf

h +−++−+

=′→

)(.)(.

)]().()(.)([)](.)()(.)([lim

0 xvhxvh

xvxuxuhxvxvxuxvhxuh +

−+−−+=

→

)(.)(.

)]()()[(])()()[(lim

0 xvhxvh

xvhxvxuxuhxuxvh +

−+−−+=

→

)(.)(

)]()([)(

])()([)(

lim0 xvhxv

h

xvhxvxu

h

xuhxuxv

h +

−+−

−+

=→

)(.)(lim

)]()([lim.)(lim

])()([lim.)(lim

0

0000

xvhxvh

xvhxvxu

h

xuhxuxv

h

hhhh

+

−+−

−+

=→

→→→→

)(.)(

)(.)(

)(.)(

xvxvdx

xdvxu

dx

xduxv −

=

��(��&��!%$

2)(

vdx

dvu

dx

duv

v

u

dx

d−

=



19

��4" .'(�#"/0��

��ก�!%$ dx

duv

dx

dvuvu

dx

d+=).( $��)"��!%$<"�

-�� ('�(�'��) = '�(�-��'�� ก '��-��'�(�

�"<��!%$ 2

)(v

dx

dvu

dx

duv

v

u

dx

d−

= ��)"��<"�

-�� (��'�� ) = � ��-�� -�� / ��-(�� ก!��/��

�� -��.�/�� )57()12( 2 +−= xxy �� dx

dy

��!� )57()12( 2 +−= xxdx

d

dx

dy → 57,12 2 +=−= xvxu

)12()57()57()12( 22 −+++−= xdx

dxx

dx

dx →

dx

duv

dx

dvuvu

dx

d+=).(

)4)(57()7)(12( 2 xxxx ++−= ,

xxxx 2028714 23 ++−= xxx 132814 23 ++= )132814( 2 ++= xxx ��(� ��-��.�/�� )57()12( 2 +−= xxy )"�ก�� )132814( 2 ++ xxx



20

�� -��.�/�� )133()4( 52 −+= xxy �� dx

dy

��!� )133()4( 52 −+=′ xxdx

dy → )133(,)4( 52 −=+= xvxu

222

1

2

122 )4()133()133()4( +−+−+= x

dx

dxx

dx

dx →

dx

duv

dx

dvuvu

dx

d+=).(

)4()4(2)133()133()133(2

1)4( 222

1

2

122 ++−+−−+=

−x

dx

dxxxx

dx

dxx ,

)2)(4(2)133()3()133()4(2

1 22

1

2

122 xxxxx +−+−+=

−

133)4(4133

)4(

2

3 222

−++−

+= xxx

x

x

��(� ��-��.�/�� )133()4( 52 −+= xxy )"�ก�� 133)4(4133

)4(

2

3 222

−++−

+xx

x

x

�� -��.�/�� 2

12

−−

=x

xy ��

dx

dy

��!� )2

1(

2

−−

=′x

x

dx

dy → 2,12 −=−= xvxu

2

22

)2(

)2()1()1()2(

−

−−−−−=

x

xdx

dxx

dx

dx

→ 2

)(v

dx

dvu

dx

duv

v

u

dx

d−

=

2

2

)2(

)1)(1()2)(2(

−−−−

=x

xxx ,

2

22

)2(142

−+−−

=x

xxx

2

2

)2(14

−+−

=x

xx

��(� ��-��.�/��212

−−

=x

xy )"�ก��

2

2

)2(14

−+−

x

xx



21

�� -��.�/�� 5

2 9

52

++

=x

xy ��

dx

dy

��!� 5

2 9

52

++

=′x

x

dx

dy →

dx

du n

++

++

=9

52

9

525 2

4

2 x

x

dx

d

x

x → 9,52 2 +=+= xvxu

+

++−++

++

=22

22

42

4

)9(

)9()52()52()9(

)9(

)52(5

x

xdx

dxx

dx

dx

x

x→

2)(

vdx

dvu

dx

duv

v

u

dx

d−

= ,

++−+

++

=22

2

42

4

)9(

)2)(52()2)(9(

)9(

)52(5

x

xxx

x

x

2242

4

22

2

42

4

)9(

)2)(52(.

)9(

)52(5

)9(

)2)(9(.

)9(

)52(5

++

++

−++

++

=x

xx

x

x

x

x

x

x

62

5

52

4

)9(

)52(10

)9(

)52(10

++

−++

=x

xx

x

x

��(� ��-��.�/��5

2 9

52

++

=x

xy )"�ก��

62

5

52

4

)9(

)52(10

)9(

)52(10

++

−++

x

xx

x

x

K�� ก��ก"� %��)��VWก�� $�� 3��VWก)�)<��< �$��



22

�� -��.�/��

1. 13

6)(

23

+−−

=x

xxxf 2. )343)(8522()( 424 −−+−−= xxxxxxf

3. 23

4)(

3

4

+−

−=

xx

xxf 4.

5

2

5

57

52)(

−−−

=xx

xxf

5. 3263 )35()53()( +−= xxxxf PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP



23




24

)�'�� ก�� 4��%

1. 992

93)( 23 +++−= xxxxf

2. x

xxxf

474)(

3 −+=

3. 32

11)(

xxxf −=

4. 5

2

3

14 59)(

−− +−= xxxxf

5. xx

xf +=2

1)(

6. 32 )2(

1

−=

xy

7. 314 xy −= 8. 22 )3( −−= xxy 9. 52 )4( −= xxy 10. )2)(78( 2 −−= xxy 11. )5)(32( 2234 −++−= xxxxxy

12. 392

+−

=x

xy

13. 92 +

=x

xy

14. xx

xxy

5

1282

2

−+−

=

15. 10

4

7

+−

=x

xy

��ก��

��%�.)�!��*



25

��ก��-�8�� (transcendental Functions) ��ก��26� �:� ��ก��)'�&3"��" ��ก��-'��12% �"� ��ก��'(ก�� ก��ก�$2)�3 ��ก��%$'�ก132%2 �� ก��%$'�ก132%2[ก[�� %� $!��/�� ก��26� 1. ��ก��'(ก�� ( The Exponential Functions) ;�� !"��$!� xay = �$:� xey = 2. ��ก��ก�$2)�3 (The Logarithmic Functions) ;�� !"��$!� xy alog= �$:� xy ln=

3. ��ก��%$'�ก132%2 (The Trigonometric Functions) ;��3' 6 ��ก�� :�

xxecxxxx cot,sec,cos,tan,cos,sin 4. ��ก��%$'�ก132%2[ก[�� (Inverse Trigonometric Functions) ;��3' 6 ��ก�� 3'

��กB1��'(

1. xy 1sin−= กg%"�3:�� xy =sin 3:�� -1 ≤ x ≤ 1, -2π≤ y ≤

2π

2. xy 1cos−= กg%"�3:�� xy =cos 3:�� -1 ≤ x ≤ 1, 0 ≤ y ≤ π

3. xy 1tan−= กg%"�3:�� xy =tan 3:�� - ∞ < x < ∞ , -2π

< y < 2π

4. xecy 1cos −= กg%"�3:�� xyec =cos 3:�� x ≥ 1, y ε [0 , 2π

) ∪ (0 , 2π

]

5. xy 1sec−= กg%"�3:�� xy =sec 3:�� x ≥ 1, y ε [0 , 2π

) ∪ (2π

, π]

6. xy 1cot−= กg%"�3:�� xy =cot 3:�� - ∞ < x < ∞ , 0 < y < π

ก��'��ก��-�8��,-�.�(/0��



26

ก��'��ก��ก��$" (Differentiation of Logarithmic Function) 3'�!%$ 2 �!%$ ��'( �� y′ 3:�� )152ln( 2 +−= xxy

<2.')�� ก dx

du

uu

dx

d 1ln =

�� )152ln( 2 +−= xxy

��(� )152ln( 2 +−=′ xxdx

dy

152

54

)152(152

1

2

22

+−−

=

+−+−

=

xx

x

xxdx

d

xx

�� y′ 3:�� )75(log 3

3 −= xy

<2.')�� ก dx

due

uu

dx

daa log

1log =

�� )75(log 33 −= xy

��(� )75(log 33 −=′ x

dx

dy

e

x

x

xdx

de

x

33

2

333

log75

15

)75(log75

1

−=

−−

=

1. dx

due

uu

dx

daa log

1log =

2. dx

du

uu

dx

d 1ln =

พิจารณาให้ดีนะครับ...ว่าควรจะใช้สูตรใดประกอบด้วย



27

%�<� "�� "��-��.�/�� ก��%"�&��'(

1. 43 )12ln()( +−= xxxf 2. )92(log

142)(

5

4

+−−

=x

xxxf

3. )726ln(.)( 3 +−= xxxxf 4.

+−

=2

13log)(

2

7 x

xxf




28




29

ก��'��ก��%ก!�� (Differentiation of Exponential Function) �� )(,)( xvxu �� ก�� 3'�!%$��'( �� -��.�/�� 252 2

4)( +−= xxxf

<2.')�� ก dx

duaaa

dx

d uu ln=

�� 252 2

4)( +−= xxxf ��(� 252 2

4)( +−=′ xxxf

4ln4)54(

)252(4ln4

252

2252

2

2

+−

+−

−=

+−=

xx

xx

x

xxdx

d

�� -��.�/�� 73)( −= xexf

<2.')�� ก dx

duee

dx

d uu =

�� 73)( −= xexf ��(� 73)( −=′ xexf

73

73

3

)73(

−

−

=

−=

x

x

e

xdx

de

�� -��.�/�� 52

)35()( +−+= xxxxf

��!� ��ก dx

dvuu

dx

duvuu

dx

d vvv ln1 += −

�� 52

)35()( +−+= xxxxf

��(� 52

)35()( +−+=′ xxxdx

dxf

)35ln()35)(12()35)(5(5

)5()35ln()35()35()35)(5(

542

25152

22

22

++−+++−=

+−++++++−=

+−+−

+−−+−

xxxxxx

xxdx

dxxx

dx

dxxx

xxxx

xxxx

dx

dvuu

dx

duvuu

dx

ddx

duee

dx

ddx

duaaa

dx

d

vvv

uu

uu

ln.3

.2

ln.1

1 +=

=

=

−



30

�� -��.�/�� 1. 3ln.)(

2

xexf x= 2. 535)( −= xxf 3. 8323 2

)23()( −−−+= xxxxxf

4. 33 )14(

)(+−

=xx

exf

x

5. xxxf ln.7)( =




31




32

ก��'��ก��,ก�"�� (Differentiation of Trigonometric Function) 3'�!%$ 6 �!%$ ��'( �� -��.�/�� )152cos()( 2 +−= xxxf

<2.')�� ก dx

duuu

dx

dsincos −=

�� )152cos()( 2 +−= xxxf

��(� )152cos()( 2 +−=′ xxdx

dxf

)152sin()54(

)152()152sin(

2

22

+−−−=

+−+−−=

xxx

xxdx

dxx

�� -��.�/�� )73tan(.)(

2

−= xexf x

<2.')�� ก dx

duv

dx

dvuvu

dx

d+=).(

�� )73tan(.)(2

−= xexf x

��(� 22

)73tan()73tan()( xx edx

dxx

dx

duexf −+−=′

)73tan(2)73(sec.3

).73tan()73()73(sec.

22

22

2

22

−+−=

−+−−=

xxexe

xdx

dexx

dx

dxe

xx

xx

1. dx

duuu

dx

dcossin =

2. dx

duuu

dx

dsincos −=

3. dx

duuu

dx

d 2sectan =

4. dx

duuecu

dx

d 2coscot −=

5. dx

duuuu

dx

dtan.secsec =

6. dx

duuecuecu

dx

dcot.coscos −=



33

�� -��.�/��

1. )(cos)( xeecxf = 2. )423cot( 4

5)( −+= xxxf 3. )8sin(

164)(

3

23

x

xxxxf

+−−=

4. xxf sec)( = 5. 32 tan.)( xxxf = PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP



34




35

ก��'��ก��,ก�"��3ก3�� (Differentiation of Trigonometric Function) 3'�!%$ ��'(

�� 3:��ก�� y = sin-1 (3x - 4) �� y′ PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP

�� 3:��ก�� y = tan-1(3x2+5)2 PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP

1. dx

d sin-1 u = dx

du

u 21

1

−

2. dx

d cos-1 u = dx

du

u21

1

−

−

3. dx

d tan-1 u = dx

du

u 21

1

+

4. dx

d csc-1 u = dx

du

uu 1

12 −

−

5. dx

d sec-1 u = dx

du

uu 1

12 −

6. dx

d cot-1 u = dx

du

u21

1

+−



36

ก��'��ก��*ก�� ,-�.�(กP�0ก,Q ( chain rule)

กP�0ก,Q K� )(ufy = �� )(xgu = �� ก��;��3'��-��.�)'� u �� x %�3

�� <��&� ��ก��$�ก�� fog /�� x ;��ก�� ))(( xgfy = �� ก��)'�3'��-��.�)'� x ��

dx

du

du

dy

dx

dy.=

��ก�!ก�;" ��&�

��ก�!%$ dx

dunuu

du

d nn 1−=

�� ก�� 452 2 −−= uuy �� 22 −= xu �� dx

dy

<2.')�� ก 452 2 −−= uuy

54

)452( 2

−=

−−=∴

u

uudu

d

du

dy

��ก 22 −= xu

x

xdu

d

dx

du

2

)2( 2

=

−=∴

กn�!ก�;" dx

du

du

dy

dx

dy.=

)134(2

)2).(584(

)2).(5)2(4(

)2).(54(

2

2

2

−=

−−=

−−=

−=

xx

xx

xx

xu

��ก��*ก��ก��,-��*��-��/0�



37

�� 1. ก�� 231 uy −= �� 223 +−= xxu �� dx

dy

2. ก�� 523 23 −+−= xxuy �� 35 += xu �� dx

dy

PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP



38

�� ก�� 6uy = �� xu 21+= �� dx

dy

PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP

NOTE …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………



39

ก��'��,-�� (Implicit Differentiation) ��ก��$2 � (Implicit Differentiation) �3� K�� ก��)'�ก�� 3ก�$;��<�3��3-��.�/�� x �� y ;��"�/�� y &3"&��ก3�� -��/�� x �$:��"�/�� x &3"&��ก3�� -��/�� y ก�"�<�:� ��ก��V�ก�� 3ก�$

0),( =yxF �"� x2 + y2 - a = 0, 3x2y - 3y2 + 2x - y2x - 5 = 0 ก�$��-��.�/�� ก�� $2 � )��&�� -��.�/��%"��/��/��3ก�$)' �ก�� x

�� K:�<"� y �� ก��/�� x $' กก�$��-��.��'(<"� ก�$��-��.�� $2 �

�� dx

dy /�� ก��%"�&��'( 042 22 =−++ yxyx




40

�� dx

dy /�� ก��%"�&��'( yxyyx 8473 324 −=−


NOTE PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP.



41

��-��/0� (Higher Derivative) ก�� ก�� )(xfy = �� K� )(xf ′ ��-��.�/�� )(xf $�$' ก )(xf ′ <"��-��.��/�� )(xf )(xf ′′ ��-��.�/�� )(xf ′ $�$' ก )(xf ′′ <"��-��.��/�� )(xf )(xf ′′′ ��-��.�/�� )(xf ′′ $�$' ก )(xf ′′′ <"��-��.��3/�� )(xf $��A��กB1�%"�&��'(�)��-��.��!� K� )(xfy = ��< ��&�

yxfdx

dy ′=′= )(

yxfdx

yd

dx

dy

dx

d ′′=′′==

)(2

2

yxfdx

y

dx

yd

dx

d d ′′′=′′′==

)(

3

3

2

2

nnn

n

n

n

yxfdx

y

dx

yd

dx

d d ===

−

−

)(1

1

�� ก�� 9234 234 +−+−= xxxxy �� 4,,, yyyy ′′′′′′ PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP



42

�� 1. ก�� 234)( xxf −= �� )(xf ′′ 2. ก�� xexxf 23)( = �� )(xf ′′′ PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP



43

1. ��<2.')��-:��-��.�/�� ก��%"�&��'( 1.1 43)37()( xxf −= 1.2 5432)( 2468 −−−−−= xxxxxxf 1.3 )26()1()( 382 xxxg −−=

1.4 2

2

3

25)(

x

xxg

−

−=

1.5 23 ln.2)( xexf x= 1.6 )5cos(3)( 2 −−= xxf 1.7 32 3

)13()( −−−= xxxxf 1.8 )3.(sec)( xxxf = 1.9 73 )3.(sin)(

2

xexf x= 1.10 xxf 7tan5)( =

2. ก�� 43)( 2 += xxxf �� )(xf ′ �� 2 �3

−+=′

→ h

xfhxfxf

h

)()()( lim

0

3. ก�� 223 35,22 xuuuy −=−−= �� dx

dy �� กn�!ก�;"

dx

du

du

dy.

4. ก�� 2)53ln()( −= xxf ��"� )(xf ′′

��WXก'�-

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