122 Chapter 4Differentiation
8In fact, developments in mathematical logic between 1920 and 1960 have led to the development of a ‘non-
standard analysis’ which involves an extension to the number system to include infinitesimals (Abraham
Robinson, Non-standard analysis, Princeton, 1996).
or
(4.33)
This is the differential of ywith respect to the variable t. Division by dtthen gives the
Chain Rule:
(4.34)
Although there have been conceptual difficulties with this type of manipulation of
differentials (infinitesimals or ‘infinitely small changes’), operations with differentials
can always be shown to duplicate methods involving finite changes (∆’s instead of d’s)
and limits.
84.13 Exercises
Section 4.2
1.Fory 1 = 1 x
3
, find (i)the change ∆yin ythat corresponds to change ∆xin x, (ii)∆y 2 ∆x.
2.Fory 1 = 1 x
3
, find
3.For the Langmuir isotherm find (i)the change ∆θin θthat corresponds to
change ∆pin p, (ii)
Section 4.3
Find the discontinuities of the following functions and state which are essential and which
removable. Sketch graphs to demonstrate your answers.
Section 4.4
Find the limits:
- lim
x
x
→ x
0
1
3
lim
x
x
x
→
0
2
lim
x
x
→ x
0
2
2
3
2
x
xx−
x
x
2
1
x+ 1
lim
∆
∆
p→ ∆p
0
θ
θ=
Kp
1 Kp
lim
∆
∆
x ∆
y
→ x
0
dy
dt
dy
dx
dx
dt
=×
dy
dy
dx
dx
dt
=×dt