Chain Rule - TeachNet Ireland
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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
© Annie Patton
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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
© Annie Patton
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5
Differentiate (3-4x) with respect to x.
Leaving Certificate 1999 Higher Level Paper 1 no 6(a)
Start clicking when you want to see the answer.
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
© Annie Patton
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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)
yu
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
© Annie Patton
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Diferentiate
1+4x with respect to x.
Leaving Certificate Higher 6c (i) paper 1 2003
Start clicking when you want to see the answer.
y= 1 4x (1 4x)
ym
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
Start clicking when you want to see the answer.
1
- 12
y=
=(1+4x)
1+4x
ym
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
© Annie Patton
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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
© Annie Patton
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x
Differentiate y=
x 1
2
3
Start clicking when you want to see the answer.
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
© Annie Patton
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Chain Rule
dy dy du
=
dx du dx
© Annie Patton