The Chemistry Maths Book, Second Edition

4.9 Successive differentiation 113

4.9 Successive differentiation

The derivative of a function can itself be differentiated if it satisfies the continuity and

smoothness conditions discussed in Sections 4.3 to 4.5. For example, the cubic

y 1 = 1 x

3

1 + 1 x

2

1 + 1 x 1 + 11

has (first) derivative

and this may be differentiated to give the second derivative, or second differential

coefficient,

Successive differentiation gives the third and fourth derivatives

and all higher derivatives are zero. Alternative notations (see Section 4.2) aref′(x),

f′′(x),f′′′(x),1=1for derivatives off(x), orDf,D

2

f,D

3

f,1=1, where D is the differential

operator.

0 Exercise 74

A polynomial of degree ncan have only up to the first nderivatives nonzero, but other

simple functions can be differentiated indefinitely. In particular, the exponential

function e

x

has all its derivatives equal to e

x

.

EXAMPLE 4.22Derivatives of sin 1 ax.

f(x) 1 = 1 sin 1 ax

f′(x) 1 = 1 a 1 cos 1 ax

f′′(x) 1 = 1 −a

2

1 sin 1 ax 1 = 1 −a

2

f(x)

In general, for the nth derivative,

fx

aax n

a

n

nn

nn

()

()

()

() cos

() si

=

−

−

−

1

1

12

2

if is odd

nnax if is evenn











d

dx

dy

dx

dy

dx

3

3

4

4

=, 60 =

d

dx

dy

dx

dy

dx

x













==+

2

2

62

dy

dx

=++ 321 xx

2