Differential Calculus-I - New Age International

10 A TEXTBOOK OF ENGINEERING MATHEMATICS–I
\ 2
D ( e cos xsin
x)
ax n
=
1 1
= sin x + ⋅ 2sin
x cos2x
2 4
1 1
= sin x + (sin3x
− sin x)
2 4
1 1
= sin x + sin3x
4 4
1
4
n
ax
D ( e
1 n
sinx)
+ D ( e
4
ax
sin3x)
1 ax 2 n /2 −1
= e [( a + 1) sin{ x+ ntan (1/ a)}
4
n / 2
−1
+ ( a + 9)
sin{ 3x
+ n tan ( 3/
a)}]
Example Example 3: 3: 3: Find the nth derivative of the following:
(i)
Solution:
Solution:
x + 3
( x −1)(
x + 2)
(i)
x + 3
y =
( x −1)(
x + 2)
By partial fractions
2
(ii)
x + 3
( x −1)(
x + 2)
A B
= +
x − 1 x + 2
Then x + 3 = Ax ( + 2) + Bx ( −1)
Taking x = 1 ⇒ A = 4/3
Equation (1), becomes
x =−2 ⇒ B =−1/3
x + 3 4 1 1 1
= ⋅ −
( x −1)(
x + 2)
3 ( x− 1) 3( x+
2)
2
x
( x + 2)(
2x
+ 3)
⎡ x + 3 ⎤
D ⎢
⎥
⎣(
x −1)(
x + 2)
⎦
n 4 n⎛ 1 ⎞ 1 n⎛
1 ⎞
= D D
3
⎜ −
x− 1
⎟
3
⎜
x+
2
⎟
⎝ ⎠ ⎝ ⎠
n n
4( −1) ⋅n! 1( −1) ⋅n!
= −
3 ( x − 1) 3(
x + 2)
n+ 1 n+
1
1 n ⎡ 4 1 ⎤
= ( −1) ⋅n! ⎢ −
3 n+ 1 n+
1⎥
⎣( x− 1) ( x+
2) ⎦
...(1)
[Using the formula (12)]