The Chemistry Maths Book, Second Edition

120 Chapter 4Differentiation

Consider the cubic

Let ∆ybe the change in yaccompanying the change ∆xin x:

∆y 1 = 1 f(x 1 + 1 ∆x) 1 − 1 f(x) 1 = 1 (x 1 + 1 ∆x)

3

1 − 1 x

3

= 13 x

2

1 ∆x 1 + 13 x(∆x)

2

1 + 1 (∆x)

3

The derivativedy 2 dxis obtained by dividing this expression by∆xand letting∆x 1 → 10.

Another way of looking at the limit is to consider ∆xas a ‘very small’ change. If∆xis

made small enough then the term in(∆x)

3

becomes much smaller than the term in

(∆x)

2

which in turn becomes much smaller than the term in ∆x,

(∆x)

3

1 << 1 (∆x)

2

1 << 1 ∆x

For example,∆x 1 = 110

− 3

,(∆x)

2

1 = 110

− 6

, and(∆x)

3

1 = 110

− 9

. An approximate expression

for the change in yis then

∆y 1 ≈ 13 x

2

1 ∆x 1 = 1 f′(x) 1 ∆x

and this is often a useful way of approximating small changes. The quantityf′(x)∆x

would be the change in yif∆xwere small enough. It is usefulto consider an arbitrary

small change dx, an ‘infinitesimal change’, such that terms in (dx)

2

and higher can be

set to zero. The corresponding change in y

dy 1 = 1 f′(x) 1 dx (4.30)

is called the differentialof y.

7

The use of the differential will become clear in later chapters. It is important in the

physical sciences because fundamental theorems are sometimes expressed in differ-

ential form; in particular, the laws of thermodynamics are nearly always expressed in

terms of differentials.

0 Exercises 89 – 91

EXAMPLE 4.26The differential area of a circle

The area of a circle as a function of the radius is

A(r) 1 = 1 πr

2

yfx x

dy

dx

==,( ) =fx x′( )=

32

3

7

Leibniz’s formulation of the calculus was in terms of differentials. His 1684 paper contains the formulas

dx

n

1 = 1 nx

n− 1

dx, for the infinitesimal change or differential ofx

n

, anddxy 1 = 1 xdy 1 + 1 ydxfor the product rule (see

Example 4.27).