using leibnitzs theorem what is nth derivative of xsinx
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Answer:
The Leibniz formula expresses the derivative on
n
th order of the product of two functions. Suppose that the functions
u
(
x
)
and
v
(
x
)
have the derivatives up to
n
th order. Consider the derivative of the product of these functions.
The first derivative is described by the well known formula:
(
u
v
)
′
=
u
′
v
+
u
v
′
.
Differentiating this expression again yields the second derivative:
(
u
v
)
′
′
=
[
(
u
v
)
′
]
′
=
(
u
′
v
+
u
v
′
)
′
=
(
u
′
v
)
′
+
(
u
v
′
)
′
=
u
′
′
v
+
u
′
v
′
+
u
′
v
′
+
u
v
′
′
=
u
′
′
v
+
2
u
′
v
′
+
u
v
′
′
.
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