Differential Calculus-I - New Age International

DIFFERENTIAL CALCULUS-I 7
dy
y 1 =
dx
2
= ( m loga)
⋅ y
d y
y 2 = = m loga ⋅ y
2
1
dx
= ( m loga)(
m loga)
⋅ y
3
= ( mloga)
d y
2
y 3 = = ( mlog a) ⋅y
3
dx
= ( mloga)
2
3
y
y
........................
........................
n
d y
y n = n
m a y
n = ( log ) ⋅
dx
n mx
[ ]
D a = ( mlog a) ⋅a
1
n mx
WORKED EXAMPLES
Example Example 1: 1: Find the nth derivative of the following functions:
Solution:
Solution:
(i) 3
y = sin x
(i) sin 3 x (ii) cos 4 x.
where 3
x
\ (sin )
3 D x
n
(ii) y = cos 4 x
where 2
x
so that 4
cos x
1
sin = ( 3sin
x − sin3x)
4
3 n 1 n
= D (sinx)
− D (sin3x)
4 4
=
3 ⎛ nπ
⎞ 3 ⎛ nπ
⎞
sin⎜
x + ⎟ − sin⎜3x
+ ⎟
4 ⎝ 2 ⎠ 4 ⎝ 2 ⎠
1
cos = ( 1 + cos2x)
2
1
= ( 1+
cos2x)
4
1 2
=
[ 1+
cos 2x
+ 2cos2x]
4
2
n
[Using the formula (15)]