494 Higher Engineering Mathematics
In general, y
(n)
= a
n sin
ax +
nπ
2
(2)
For example, if
y = sin 3x, then
d 5 y
dx 5 = y
(5)
= 3
5 sin
3x +
5π
2
= 3
5 sin
3x +
π
2
= 243 cos 3x
(iii) If y = cos ax,
y
=−a sin ax = a cos
ax +
π
2
y
=−a
2 cos ax = a
2 cos
ax +
2π
2
y
= a
3 sin ax = a
3 cos
ax +
3π
2
and so on.
In general, y
(n)
= a
n cos
ax +
nπ
2
(3)
For example, if y = 4 cos 2x,
then
d 6 y
dx 6 = y (6) = 4(2 6 ) cos
2x +
6π
2
= 4(2 6 ) cos (2x + 3π)
= 4(2 6 ) cos (2x + π)
= −256 cos 2x
(iv) If y = x a , y = a x a−1 , y = a(a − 1)x a−2 ,
y
= a(a − 1)(a − 2)x
a−3 ,
and y (n) = a(a − 1)(a − 2)... ..(a − n + 1) x a−n
or y
(n)
=
a!
(a − n)!
x
a−n
(4)
where a is a positive integer.
For example, if y = 2x
6 , then
d 4 y
dx 4 = y
(4)
= (2)
6!
(6 − 4)!
x 6−4
= (2)
6 × 5 × 4 × 3 × 2 × 1
2 × 1
x 2
= 720x 2
(v) If y = sinh ax, y = a cosh ax
y
= a
2 sinh ax
y
= a
3 cosh ax, and so on
Since sinh ax is not periodic (see graph on page
43), it is more difficult to find a general statement for y (n) . However, this is achieved with the
following general series:
y
(n)
=
a n
2
{[1 +(−1)
n ]sinh ax
+ [1 −(−1)
n ]cosh ax}
(5)
For example, if
y = sinh 2x, then
d 5 y
dx 5 = y
(5)
=
2 5
2
{[1 + (−1)
5 ] sinh2x
+ [1 − (−1)
5 ] cosh 2x}
=
2 5
2
{[0] sinh 2x + [2] cosh 2x}
= 32 cosh 2x
(vi) If y = cosh ax,
y
= a sinh ax
y
= a
2 cosh ax
y
= a
3 sinh ax, and so on
Since cosh ax is not periodic (see graph on page
43), again it is more difficult to find a general
statement for y (n) . However, this is achieved with
the following general series:
y
(n)
=
a n
2
{[1 − (−1)
n ]sinh ax
+ [1 + (−1)
n ]cosh ax}
(6)
For example, if y =
1
9
cosh 3x,
then
d 7 y
dx 7 = y (7) =
1
9
3 7
2
(2 sinh3x)
= 243 sinh 3x
(vii) If y = ln ax, y =
1
x
, y = −
1
x 2 , y =
2
x 3 , and so
on.
In general, y
(n)
= (−1)
n−1 (n − 1)!
x n
(7)
For example, if y = ln 5x, then
d 6 y
dx 6 = y (6) = (−1) 6−1
5!
x 6
= −
120
x 6