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