Transcript General Rule - TeachNet Ireland

General Rule
© Annie Patton
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Aim of Lesson
To introduce the idea of
differentiating by Rule rather than
First Principles.
© Annie Patton
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Note
If an examination question
does not state by first principles
it should be done by some of
the rule methods, that you are
about to learn.
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Note another name for the
General Rule is the Power
Rule.
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To prove the differentiation of a
constant equals zero.
dy
For example if y=5, then
=0.
dx
Let y=f(x)=c
f(x+h)=c
f(x+h)-f(x)=c-c=0
f(x+h)-f(x) 0
lim
= =0
h 0
h
h
© Annie Patton
d(c)
=0
dx
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General Rule
dy
n-1
If y=x . Then
=nx .
dx
n
dy
2
For example if y=x . Then
=3x
dx
3
Proof of this rule is left to the Miscellaneous
Presentation, as it demands the Product Rule,
which we have not covered yet.
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Differentiate x with respect to x by rule.
Start clicking when you want to see the answer.
y= x =x
1
2
1
2
dy 1
1
= x =
dx 2
2 x
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df(x)
dg(x)
Proof that if f(x)=kg(x), where k is a constant. Then
=k
.
dx
dx
2
dy
dx
2
1
For example y=5x ,
=5
=5(2x )=10x
dx
dx
f(x)=kg(x)
f(x+h)=kg(x+h)
f(x+h)-f(x) (g(x+h)-g(x))
=k
h
h
f(x+h)-f(x)
(g(x+h)-g(x))
lim
=k lim
h 0
h 0
h
h
df(x) dg(x)
=k
dx
dx
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Differentiate y=5x 4 with respect to x.
Start clicking when you want to see the answer.
d(5x 4 )
d(x 4 )
3
3
=5
=5(4x )=20x
dx
dx
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5
Differentiate 3 with respect to x.
x
Start clicking when you want to see the answer.
5
-3
y= 3 =5x
x
dy
15
-4
=-15x =- 4
dx
x
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The Sum Rule
If f(x)=u(x)+v(x)
f(x+h)=u(x+h)+v(x+h)
f(x+h)-f(x)=u(x+h)+v(x+h)-u(x)-v(x)
f(x+h)-f(x)  u(x+h)-u(x)   v(x+h)-v(x) 
=
+


h
h
h

 
f(x+h)-f(x)
 u(x+h)-u(x) 
 v(x+h)-v(x) 
= lim 
+
lim
 h 0 

h 0
h 0
h
h
h

lim
df(x) d(u(x)+v(x)) du(x) dv(x)
=
=
+
dx
dx
dx
dx
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3
2
Differentiate y=x +6x +5x+3 with respect to x.
Start clicking when you want to see the answer.
dy
2
=3x +12x+5
dx
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dy
If y  x  3x  4, find
when x  2.
dx
4
2
Start clicking when you want to see the answer.
dy
3
=4x +6x
dx
dy
At x=2,
=4(2)3 +6(2)=32+12=44
dx
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Differentiate the following with
respect to x:
1. y=6x
4
3
3
2. y=x +2x +8
1 1
3. y= 2 +
x x
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