Differential Calculus-I - New Age International

DIFFERENTIAL CALCULUS-I 3
3
d y
3 m−2
y3
= = m(
m −1)
( m − 2)
a ( ax + b)
3
dx
........................................................
........................................................
yn
n m−n
( m −1)
( m − 2)
L ( m − n + 1)
a ( ax + b
...(5)
= m
)
This formula is true for all m.
Following are some particular cases.
Case Case ( (iiiii): ( ( ): Suppose m = n (a +ve integer)
In equation (5) becomes,
n
n
n n n
D [( ax + b)
] = nn ( 1) ( n 2) 1 a( ax b) −
− − ⋯ ⋅ +
n
= n! a
...(6)
In particular,
( )
n n D x = n!
...(7)
Case Case (
( (ii ii): ii ii ii ): Suppose m is a positive integer and m > n. Then formula (5) becomes
n
D [( ax + b)
m
]
m( m−1) …( m− n+ 1)( m−n)( m−n−1) …2⋅1
= ⋅ a ( ax+ b)
( m−n)( m−n−1) …2⋅1
m! n m − n
= a ( ax + b)
( m − n)
!
In particular,
( )
m n m! m − n
D x = x
( m − n)
!
Case Case ( (
( (iii iii): iii iii iii ): Suppose m is a positive integer and n > m. From (6) we note that
n m−n ...(8)
...(9)
n m
D [( ax + b)
] ! m
= m a
In differentiate further, the right-hand side gives zero.
Thus, n m
D [( ax + b)
] = 0 if n > m ...(10)
In particular,
( )
m n D x = 0 for n > m ...(11)
Case Case ( (
( (iv iv): iv iv iv ): Suppose m = – 1, in this case formula (5) becomes,
n ⎡ 1 ⎤
n
D ⎢ax + b⎥
= ( −1)
( −2)
( −3)
K(
−n)
a ( ax + b)
⎣ ⎦
n n
( 1) n! a
( ax b )
+
− ⋅
=
+
n 1
−1−n
...(12)