
Successive Differentiation 5-11
Example 5.15 If y ax b= +sin( ), then show that y a ax b n
n
n
= + +sin( [ / ]).p 2
Solution: Given that y ax b= +sin( ).
y a ax b
a ax b
y a ax b
a a
1
2
2
2
= +
= + +
= + +
=
cos( )
sin( [ )
cos( [ )
sin(
p
p
/2]
/2]
xx b
a ax b
y a ax b
+ + +
= + + ⋅
= + + ⋅
[ [ )
sin( [ ])
sin( [
p p
p
p
/2] /2]
/2
/2]
2
3
3
2
3 ))
sin( [ / ]).
∴ = + +y a ax b n
n
n
p 2
Similarly, we can prove if
y ax b= +cos( ) then y a ax b n
n
n
= + +cos( [ / ]).p 2
Example 5.16 Find the nth derivative of sin ax cos bx.
Solution:
d
dx
ax bx
d
dx
ax bx
d
dx
n
n
n
n
n
n
[sin cos ] [ sin cos ]
si
=
=
1
2
2
1
2
nn( ) sin( )
sin( ) sin(
a b x a b x
d
dx
a b x
d
dx
a b
n
n
n
n
+ + −
{ }
= + + −
1
2
))
( )