12a: The Quotient Rule 12a: The Quotient Rule © Christine Crisp “ “ Teach A Level Maths” Teach A Level Maths” Vol. 2: A2 Core Vol. 2: A2 Core Modules Modules
Dec 17, 2015
12a: The Quotient Rule12a: The Quotient Rule
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules
The Quotient Rule
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Module C3
OCR
The Quotient RuleThe following are examples of
quotients:
x
ey
x
1
(b)2
21
x
xy
(a)
(a) can be divided out to form a simple function as there is a single polynomial term in the denominator.
2
21
x
xy
For (b) we use the quotient rule.
2
2
2
1
x
x
xy
12 xy
32 xdx
dy
The Quotient RuleThe quotient rule gives us a way of
differentiating functions which are divided.
2vdx
dvu
dx
duv
dx
dy
v
uy
The rule is similar to the product rule.
where u and v are functions of x.Memory aid:
2bottom bottom grad x top – top grad x Bottom
The Quotient Rule
2vdxdv
udxdu
v
dx
dy
v
uy
2v
vuuv
dx
dy
v
uy
Or using the dash notation
2bottom bottom grad x top – top grad x Bottom
The Quotient Rule
1
2
x
xye.g. 1 Differentiate to find .
dx
dy
2vvu-uv
2v
dxdv
udxdu
v
dx
dy
v
uy
xdx
duu 2 1
dx
dvv
We now need to simplify.
v
uySolutio
n: 1 xvand
2
2
1
112
)()(
x
xxx
dx
dy
2xu
The Quotient Rule
2
2
)1(
)1(2
x
xxx
dx
dy
2
22
)1(
22
x
xxx
dx
dy
2
2
)1(
2
x
xx
dx
dy
2)1(
)2(
x
xx
dx
dy
We could simplify the numerator by taking out the common factor x, but it’s easier to multiply out the brackets. We don’t touch the denominator.
Now collect like terms:
and factorise:
We leave the brackets in the denominator as the factorised form is simpler.
Multiplying out numerator:
The Quotient RuleQuotients can always be turned into
products.
However, differentiation is usually more awkward if we do this.
21 x
e x
e.g. 12 )1( xe xcan be written
as
In the quotient above, and
xeu 21 xv
In the product , andxeu 12 )1( xv
( both simple functions )
( v needs the chain rule )
The Quotient RuleSUMMARY
2vdx
dvu
dx
duv
dx
dy
Otherwise use the quotient rule:If ,
v
uy
where u and v are both functions
of x
To differentiate a quotient:
Check if it is possible to divide out. If so, do it and differentiate each term.
The Quotient RuleExercis
eUse the quotient rule, where appropriate, to differentiate the following. Try to simplify your answers:
xe
xy
3
1.
2.
x
xy
2
2
3.
2
2
x
xy
The Quotient Rule
xe
xy
3
1.
23xdx
duu xe
dx
dvv
2vdxdv
udxdu
v
dx
dy
v
uy
and3xu xev
v
uy
2
323
)( x
xx
e
exex
dx
dy
2
2 3
)()(
x
x
e
xex
dx
dy xe
xx
dx
dy )(
32
Solution:
The Quotient Rule
2.
xdx
duu 2 1
dx
dvv
2vdx
dvu
dx
duv
dx
dy
v
uy
and2xu xv 2v
uy
2
2
2
122
)()()(
x
xxx
dx
dy
Solution:
x
xy
2
2
2
22
2
24
)( x
xxx
dx
dy
2
2
2
4
)( x
xx
22
4
)()(
x
xx
The Quotient Rule
22
2
x
x
xy
2
2
x
xy
234 xxdx
dy
Solution:
3.
2
2
x
xy
Divide out:
122 xxy
1
23
14
xx
The Quotient Rule
The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
The Quotient RuleThe following are examples of
quotients:
x
ey
x
1
(b)2
21
x
xy
(a)
(a) can be divided out to form a simple function as there is a single polynomial term in the denominator.
2
21
x
xy
For (b) we use the quotient rule.
2
2
2
1
x
x
xy
12 xy
32 xdx
dy
The Quotient Rule
2vdx
dvu
dx
duv
dx
dy
SUMMARY
Otherwise use the quotient rule:If ,
v
uy
where u and v are both functions
of x
To differentiate a quotient:
Check if it is possible to divide out. If so, do it and differentiate each term.
The Quotient Rule
12 xvxuv
uy and
Solution:
1
2
x
xye.g. 1 Differentiate to find .
dx
dy
2vdx
dvu
dx
duv
dx
dy
v
uy
xdx
du2 1
dx
dv
2
2
)1(
)1(2
x
xxx
dx
dy
We now need to simplify.
The Quotient Rule
2
2
)1(
)1(2
x
xxx
dx
dy
2
22
)1(
22
x
xxx
dx
dy
2
2
)1(
2
x
xx
dx
dy
2)1(
)2(
x
xx
dx
dy
We could simplify the numerator by taking out the common factor x, but it’s easier to multiply out the brackets. We don’t touch the denominator.
Now collect like terms:
and factorise:We leave the brackets in the denominator. ( A factorised form is considered to be simpler. )
Multiplying out numerator: