The Chemistry Maths Book, Second Edition

(Grace) #1

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
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