Transcript Chain Rule - TeachNet Ireland

Chain Rule
© Annie Patton
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Aim of Lesson
To demonstrate the workings and use
of the Chain Rule.
© Annie Patton
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Chain Rule
This rule is used to differentiate composite functions.
For example y=(3x+2)
© Annie Patton
4
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Chain Rule method
n
If y=u , where u is
expressed in terms x.
Then
dy dy du
=
dx du dx
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4
Differentiate y=(3x+2)
du
y=(3x+2) , then u=3x+2 and
3
dx
4
y=u 4
dy
3
3
=4u  4(3x  2)
du
dy dy du
=
=4(3x+2)3 3=12(3x+2)3
dx du dx
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5
Differentiate (3-4x) with respect to x.
Leaving Certificate 1999 Higher Level Paper 1 no 6(a)
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Let u=3 - 4x
du
=-4
dx
y=(3-4x)5 =u 5
dy
=5u 4 =5(3-4x) 4
du
dy dy du
=
=5(3 - 4x) 4 .(- 4)= - 20(3 - 4x) 4
dx du dx
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2
Find the derivative of x +1
Leaving Certificate 1999 Higher Level Paper 1 no 7(a)
Start clicking when you want to see the answer.
2
1
2
1
2
1
y=(x +1)
yu
1
2
dy 1
 u
du 2
-

2 x 1
2
Let u=x 2 +1
du
=2x
dx
dy dy du
1
x
=
=
.2x=
2
dx du dx 2 x +1
x 2 +1
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Diferentiate
1+4x with respect to x.
Leaving Certificate Higher 6c (i) paper 1 2003
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y= 1  4x  (1  4x)
ym
1
2
Let m=1+4x
dm
4
dx
1
2
dy
1  12
1
1
 m 

dm 2
2 m
2 1 4x
dy dy dm
1
2


4
dx dm dx 2 1  4 x
1 4x
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Diferentiate
1
with respect to x.
1+4x
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1
- 12
y=
=(1+4x)
1+4x
ym

Let m=1+4x
dm
4
dx
1
2
dy
1  32
1
1
 m 

3
dm
2
2( m )
2( 1  4 x )3
dy dy dm
1
2


4
3
dx dm dx
2( 1  4 x )
( 1  4 x )3
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5
 2x 
Differentiate y=  2  with respect to x , when x  1.
 x 1 
Start clicking when you want to see the answer.
2x
Let u= 2
x +1
du (x +1)(2)-2x(2x) 2x +2-4x
2  2x
=
=
 2
2
2
2
2
dx
(x +1)
(x +1)
( x  1)2
2
y=u 5
dy
 2x 
4
=5u =5  2 
du
 x +1 
4
2
dy dy du  2x 
=
=5  2 
dx du dx  x +1 
2
4
2
 2 -2x 2 
 2 2
 (x +1) 
dy  2 
When x=1,then
=5   (0)=0
dx  2 
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 x 
Differentiate y= 

 x 1
2
3
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x2
m
x 1
du
=2x
dx
y  m3
2


dy
x
2
 3m  3

dm
(
x

1


v  x 1
dv
1
dx
u=x 2
2
dm ( x  1)(2x)  x2 (1) 2x2  2x  x2 x2  2x



2
2
dx
( x  1)
( x  1)
( x  1)2
2
 x   x2  2x 
dy dy dm

 3
 
2 
dx dm dx
x

1
(
x

1)

 

2
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Differentiate the following exercises
with respect to x:
1. (1+7x)3 Leaving Certificate Higher paper1 2005 no6(a)(i)
2. (x 4  1)5 Leaving Certificate Higher paper1 2002 no6(a)(i)
dy
3. If y= 3 x  5, what is ?
dx
 2x  5 
4. If y=  2  , what is
 x 
 1 
5. If y= 
 , what is
 2x  5 
4
dy
?
dx
dy
?
dx
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Chain Rule
dy dy du
=
dx du dx
© Annie Patton