3.4 連鎖法則及應用 連鎖法則及應用 連鎖法則及應用 連鎖法則及應用—相關變率 相關變率 相關變率 相關變率
利用連鎖法則求下列 利用連鎖法則求下列 利用連鎖法則求下列
利用連鎖法則求下列 1-3 題之題之題之題之
111 1 xxx
dx
xdx dx dx dy dy dy dy dx dx dx dx dy dy dy dy
=
與與 與與
1. y=u2 −3u+5,u= x3 −2x2 +2
) 4 3 ( ) 3 2 ( ) 2 2 ( 5) 3 -
( 2 x3 x2 u x2 x
dx u d
du u d dx du du dy dx
dy = ⋅ = + ⋅ − + = − ⋅ −
1 ) 1 ( ) 1 ( ) 4 3 ( ) 3 1 2 (
1 2 1 2 1
1
1
2 3
=
−
⋅
−
=
−
⋅
−
⋅
=
∴
= +
⋅
−
=
= ⇒
=
dx x
dy u
x ，代入
2. y= 2u2 +1,u=2x2 +x−1
) 1 4 ( 1 2 ) 2 1 4 ( 1 2 2 ) 4 1 2
( ) 1 2
( 2 2
2
2 ⋅ +
= + + + ⋅
=
− +
⋅ +
=
⋅
= x
u x u
u x u
dx x u du
du d dx du du dy dx dy
3 5 20 3 ) 4 1 1 4 ( 1 2 2
2 2
2 1 1 1 2
1
1 2
2
=
⋅
= +
⋅ + ⋅
⋅
= ⋅
∴
=
− +
⋅
=
= ⇒
=
dx x
dy u
x ，代入
3. , 3 1
1
2 −
− =
= u x
u y u
2 2
2
) 1 ( 6 6 ) 1 (
) 1 ) (
1 3 ( 1)
( −
= −
− ⋅
−
= −
−
− ⋅
=
⋅
= u
x x u
u x u
dx d u
u du
d dx du du dy dx dy
) 6 1 2 (
1 6
2 1 1 3
1
2 1
2
−
− =
⋅
= −
∴
=
−
⋅
=
= ⇒
=
dx x
dy u
x ，代入
3.4 連鎖法則及應用 連鎖法則及應用 連鎖法則及應用 連鎖法則及應用
求下列各函數之導函數 求下列各函數之導函數 求下列各函數之導函數 求下列各函數之導函數(4-13)
4. y=(x2 +5x−2)4
) 5 2 ( ) 2 5 ( 4 ) 2 5 ( ) 2 5 ( 4 ) 2 5
( 2 + − 4 = 2 + − 3⋅ 2 + − = 2 + − 3⋅ +
′= x x x x x
dx x d
x x
dx x y d
5. 3
1 3
3 3
) 3 (
3x y x x
x
y= − ⇒ = −
3 3 2
2 3 2
2 3
3 3 2 3 3
1 3
) 3 ( ) 1 3 3 ( ) 3 3(
) 1 3 ( )
3 3(
) 1 3 (
x x x x
x x x
dx x x d x x
dx x y d
−
= −
−
⋅
−
=
−
⋅
−
=
−
′= − −
6. 2 3 5( 2 4) 3 )
4 (
5 −
+
= + ⇒
= y x
y x
4 2 4
2 2
4 2 3
2
) 4 ( 2 30 ) 4 ( 15 ) 4 ( )
4 ( 15 ] ) 4 ( 5
[ +
= −
⋅ +
−
= +
⋅ +
−
= +
′= − − −
x x x x
dx x x d
dx x y d
7.
3 2
2
1
−
= x x y
解法一：
( )
( ) ( ) ( ) ( )
7 2 3
4 2 3
2 2
3 2 2 2 3
2 2 2 1 2
2 1 2 1
2 3 1 3
2
) 4 ( ) 2 ( 3 ) 4 ( ) 2 3 (
4 3 2
4 1 2
3 1 4
2 3
2 2
3 2 2 1
x x x
x x x
x x
x x
x
x x x x x
x x
x x
dx x x d
x y
x x x y
y x
−
−
= − +
⋅ −
⋅ −
=
− +
−
=
− +
−
= +
−
−
=
−
⋅
−
′=
−
=
⇒
−
=
−
−
−
−
−
−
−
−
−
−
解法二：利用除法公式
3 2 3
2
2 2
1
−
=
⇒
−
= x
y x x
y x
7 2 3
4 2
4 4
2 4
2 4
2 4
2 4
2
2 2
2 2
2 2 2
2 2 3
2
) 4 ( ) 2 ( 3 ) 4 ( ) 2 3 (
) 4 ( )
2 3 (
) 4 (
) 2 3 (
2 ) 2 ( 1 )
2 3 (
) (
) ( ) 2 ( ) 2 2 (
2 3 3 2
2
x x x
x x x
x
x x x x
x x
x x x
x x
x x
x x
x
x
dx x x d
dx x x d
x x x
x dx
d x
x x
x dx y d
−
−
= −
−
⋅−
⋅ −
=
−
⋅−
⋅ − + =
⋅ −
⋅ −
⋅ =
−
−
⋅ ⋅
⋅ −
=
⋅
−
−
−
⋅
⋅
−
⋅
=
−
⋅
−
=
−
′=
8. 2
1 2
2 2
2 + −3⇒ = ( + −3)
=x x x y x x x
y
3 2
) 1 2 3 (
2
) 1 2 ( ) 3 2 (
) 1 3 (
2
) 3 (
) 3 2 (
2 1 ) 3 (
) 3 (
) ( ) 3 (
) 3 (
2 2 2
2 1 2
2 2 1 2
2 2 1 2
2 2 1 2
2 1 2
2 2 2
1 2 2
1 2
2
− + + +
− +
=
+
⋅
− +
⋅
⋅ +
− +
=
− +
⋅
− +
⋅
⋅ +
⋅
− +
=
− +
⋅ +
⋅
− +
=
+ −
′=
−
−
x x
x x x
x x
x x
x x
x x x
x dx x
x d x x
x x
x
x dx x
x d dx x
x d x x
x dx x y d
9. 2 4
) 3 (
6
= − x y x
[ ]
5 2
2 5
2
2 2
5 2
2 4
2 8
2
3 2 2 4
2 8
2
3 2 4
2
8 2
2 3
2 4
2
4 2 2
4 2 4
2
) 3 (
18 42
) 3 (
48 ) 3 6(x
) 3 (
48 )
3 (
6 )
3 (
) 3 ( 48 ) 3 ( 6 )
3 (
2 ) 3 ( 4 6 6 ) 3 (
) 3 (
) 3 dx(
) d 3 ( 4 6 6 ) 3 ( )
3 (
3) - (x 6 ) 6 ( ) 3 (
−
−
= −
−
−
= −
− −
= −
−
−
−
= −
−
⋅
−
⋅
−
⋅
= −
−
−
⋅
−
⋅
−
⋅
= −
−
−
= −
′
x x x
x
x x x
x
x x x
x
x x
x x
x
x x
x x
x
dx x d dx x
x d y
10. y=(x2 +1)4(x3 −2x+1)3
) 2 3 ( ) 1 2 ( ) 1 ( 3 ) 1 ( ) 1 2 ( 8
) 2 3 ( ) 1 2 ( 3 ) 1 ( ) 2 ( ) 1 ( 4 ) 1 2 (
) 1 2 ( ) 1 2 ( 3 ) 1 ( ) 1 ( ) 1 ( 4 ) 1 2 (
) 1 2 ( ) 1 ( ) 1 ( )
1 2 (
2 2 3
4 2 3 2 3 3
2 2 3
4 2 3
2 3 3
3 2
3 4 2 2
3 2 3 3
3 3
4 2 4 2 3
3
− +
− +
+ + +
−
=
−
⋅ +
−
⋅ + +
⋅ +
⋅ +
−
=
+
−
⋅ +
−
⋅ + + +
⋅ +
⋅ +
−
=
+
−
⋅ +
+
+
⋅ +
−
′=
x x
x x
x x
x x
x x
x x
x x
x x
x dx x
x d x x
dx x x d
x x
x dx x
x d dx x
x d x y
11.
2 1 2 1 2
2 2 4 (2 4)
+ +
= + ⇒
+
= x x y x x
y
4 2 2 1 4 2 2
1
) 4 2 ( 2x )
4 2 2 (
2 1 ) 4 2 2( 2x 1 )
4 2 2 (
1
) 4 2 ( )
4 2 2( 2x 1 )
4 2 2 (
) 1 4 2 ( )
4 2 2 (
1
2
2 2 1
1 2 1 2 2
2 1 1 2 1 2
2 2 1
1 2 1 2 2
1 2
2 1 2 1 2
+ + +
= +
+ +
⋅
+ +
=
+ + ⋅
⋅
+ +
=
+ + ⋅ +
⋅
+ +
=
+ +
⋅
+ +
′=
− −
− −
− −
−
x x x
x
x x
x x
x x
dx x x d
x x
x dx x
x d x
y
12.
2 1
2 2 2
2
1 1 1
1
−
= +
− ⇒
= +
x y x x
y x
2 2 2
2 2
2 2 1
2 2 2
2 2 2 2
1
2 2
2 2
2 2
2 2
2 1
2 2 2
2 2 1
2 2
) 1 (
2 1
1 )
1 (
4 1
1 2
1 )
1 (
2 ) 1 ( 2 ) 1 ( 1 1 2
1
) 1 (
) 1 ( ) 1 ( ) 1 ( ) 1 ( 1 1 2
1 1 1 1
1 2
1
−
⋅ − +
= −
−
⋅ −
−
= +
−
⋅ +
−
⋅
⋅ −
−
= +
−
−
⋅ +
− +
⋅
⋅ −
−
= +
−
⋅ +
−
= +
′
−
−
−
−
x x x
x x
x x
x x
x x
x x
x x
x
dx x x d
dx x x d
x x x
x dx
d x
y x
13. y=(3x2 +(x3 −2x+1)3)4
[ ] [ ]
[ ]
[
3 ( 2 1)] [
6 3( 2 1) (3 2)]
4
) 1 2 ( ) 1 2 ( 3 6 ) 1 2 ( 3 4
) 1 2 ( 3 )
1 2 ( 3 4
2 2 3 3
3 3
2
3 2
3 3 3 3
2
3 3
3 2 3 3
2
− +
− +
⋅ +
− +
=
+ − + ⋅ − +
⋅ +
− +
=
+
− +
⋅ +
− +
′=
x x
x x x
x x
x dx x
x d x x x
x x
x x dx x
x d x x y
