27: Harder Differentiation - Differentiating with Negative and Rational Indices © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

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


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


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


ExercisesDifferentiate the following: 12 xxy

232 xxdx

dy

1.

23

21

xxy

21

21

2

3

2

1xx

dx

dy

2.

Ans:

Ans:


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:


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


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


ExercisesDifferentiate the following:

13 xxy

223 xxdx

dy2

2 13

xx

1.

xxxy

12323

234 49 xxxdx

dy234

149

xxx

123 23 xxxy2.


Another rule of indices enables us to differentiate expressions containing roots such as x

naan1


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


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


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



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.


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


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


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. 2 Differentiate xy

xdx

dy

2

1

nn

aa

1Usin

g

Using

nxxn1

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27: Harder Differentiation - Differentiating with Negative and Rational Indices © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules