2. Harder Differentiation
"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with
permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
Module C1
AQAEdexcel
OCR MEI/OCR
Module C2
3. Harder Differentiation
We have differentiated terms of the form
where n is a positive integer.
n
ax
e.g.
23
124 x
dx
dy
xy =⇒=
The Rule for Differentiation
The same rule holds when n is negative or a fraction.
1−
=⇒= nn
nax
dx
dy
axy
8. Harder Differentiation
32
21
1
xxdx
dy
−−=
This answer can be left like this or written as
e.g. 2 Differentiate 2
11
xx
xy ++=
Solution: 2
11
xx
xy ++=
=
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!
+−1
x 2−
x
1 2−
− x 3
2 −
− x
11. Harder Differentiation
2
1
2
1 −
= x
dx
dy
⇒
xy = ⇒
Solution:
2
1
xy =
This answer can be left like this or:
2
1
2
1
x
dx
dy
=
e.g. 1 Differentiate xy =
xdx
dy
2
1
=
n
n
a
a
−
=
1
Using
⇒
Using n
xxn
1
=
12. Harder Differentiation
⇒ =y
x
xy
1
2 +=Solution:
2
1
2
1
2
−
+ xx
3
2
11
xx
−
2
1
2
1
1
2
x
x +
⇒ =y
2
3
2
1
2
1
2
1
2
−−
−× xx⇒ =
dx
dy
⇒ =
dx
dy
e.g. 2 Differentiate
x
xy
1
2 +=
We can leave
the answer in
either form
16. 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.
20. Harder Differentiation
2
1
2
1 −
= x
dx
dy
⇒
xy =
Solution:
2
1
xy =
This answer can be left like this or:
2
1
2
1
x
dx
dy
=
e.g. 2 Differentiate xy =
xdx
dy
2
1
=
n
n
a
a
−
=
1
Using
⇒
Using n
xxn
1
=
⇒