27: harder differentiation - differentiating with negative and rational indices © christine crisp...
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27: Harder Differentiation 27: Harder Differentiation - Differentiating with - Differentiating with Negative and Rational Negative and Rational
IndicesIndices
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
Harder Differentiation
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Module C1
AQAEdexcel
OCR MEI/OCR
Module C2
Harder Differentiation
We have differentiated terms of the form where n is a positive integer.
nax
e.g.23 124 x
dx
dyxy
The Rule for Differentiation
The same rule holds when n is negative or a fraction.
1 nn naxdx
dyaxy
Harder Differentiation
e.g. 2 Find the gradient function, if )(/ xf
21
4)( xxf
21
4)( xxf
Solution:
21
2
14)(/ xxf
21
2)(/ xxf
e.g. 1 43 124 xdx
dyxy
N.B. 3 1 4
Harder Differentiation
ExercisesDifferentiate the following: 12 xxy
232 xxdx
dy
1.
23
21
xxy
21
21
2
3
2
1xx
dx
dy
2.
Ans:
Ans:
Harder Differentiation
nn
aa
1
To differentiate a term like we need to change
it to a constant multiplied by the variable.
2
4
x
We use one of the laws of indices:
Harder Differentiation
nn
aa
1
e.g.1 Find the gradient function of 2
4
xy
2
4
xy
Solution:
2
14x
y
24 xy
3)2(4 xdx
dy
38 x
Harder Differentiation
32
211
xxdx
dy
This answer can be left like this or written as
e.g. 2 Differentiate 2
11
xxxy
Solution: 2
11
xxxy
dx
dy
xy
We don’t start to differentiate until all the terms are in the right form
Only the x has a negative index so the 2 doesn’t move!
1x 2x
1 2 x 32 x
Harder Differentiation
ExercisesDifferentiate the following:
13 xxy
223 xxdx
dy2
2 13
xx
1.
xxxy
12323
234 49 xxxdx
dy234
149
xxx
123 23 xxxy2.
Harder Differentiation
Another rule of indices enables us to differentiate expressions containing roots such as x
naan1
Harder Differentiation
21
2
1 x
dx
dy
xy
Solution:
21
xy
This answer can be left like this or:
21
2
1
xdx
dy
e.g. 1 Differentiate xy
xdx
dy
2
1
nn
aa
1Usin
g
Using
nxxn1
Harder Differentiation
yx
xy1
2 Solution:
21
21
2
xx
32
11
xx
21
21 1
2x
x
y
23
21
2
1
2
12
xx
dx
dy
dx
dy
e.g. 2 Differentiatex
xy1
2
We can leave the answer in
either form
Harder DifferentiationSUMMAR
Y The rule for differentiating can be used for
nax
nx
a• using nn
xx
1
• usingn xa nxxn1
Harder Differentiation
ExercisesDifferentiate the following:
23
21
xxy 21
21
2
3
2
1xx
dx
dy
1.
xxy
22
23
21
xxdx
dy3
11
xx
21
21
22
xxy
Harder Differentiation
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.
Harder Differentiation
e.g. 2 Find the gradient function, if )(/ xf
21
4)( xxf
21
4)( xxf
Solution:
21
2
14)(/ xxf
21
2)(/ xxf
e.g. 1 43 124 xdx
dyxy
N.B. 3 1 4
Harder DifferentiationSUMMAR
Y The rule for differentiating can be used for
nax
nx
a• using nn
xx
1
• usingn xa nxxn1
Harder Differentiation
nn
aa
1
e.g.1 Find the gradient function of 2
4
xy
2
4
xy
Solution:
2
14x
y
24 xy
3)2(4 xdx
dy
38 x