Differential Calculus-I - New Age International

2 A TEXTBOOK OF ENGINEERING MATHEMATICS–I
Thus, we have the formula
In particular,
( )
ax n D e
( )
x n D e
n ax
= a e
...(1)
x
= e
...(2)
(2) (2) nth derivative of log (ax + b)
Let y = log( ax + b).
Then we find, by successive differentiation
y 1
dy a
= =
dx ax + b
2
2
d y a
y2 = = ( −1)
2
2
dx
( ax + b)
3
3
d y
a
y3 = = ( −1)
( −2)
⋅
3
3
dx
( ax + b)
4 4
d y a
y4 = = ( −1)( −2)( −3)
4 4
dx ( ax + b)
... ............................................
... ............................................
yn
Thus, we have the formula
In particular,
D n
n
d y
= = ( −1)
n
dx
n−1
n−1n ( −1) ⋅( n−1)! a
[log( ax + b)]
=
n
( ax + b)
D [logx]
n
(3) (3) nth derivative of (ax + b) m
n−1
( −1) ⋅( n −1)!
=
n
x
Let y m
= ( ax + b)
Differentiating successively, we get
y1
y2
=
dy
= ma ( ax + b)
dx
2
( n −1)!
a
⋅
( ax + b)
m−1
d
y
2
= = m(
m −1)
⋅ a ( ax + b)
2
dx
n
n
m−2
...(3)
...(4)