N order differentiation of sin x
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1
Answer:
Answer:
y
(
n
)
=
n
sin
(
x
+
(
n
−
1
)
π
2
)
+
x
sin
(
x
+
n
π
2
)
Explanation:
We seek the
n
t
h
derivative of:
f
(
x
)
=
x
sin
x
Starting with the given function:
f
(
0
)
(
x
)
=
x
sin
x
Using the product rule we compute the first derivative:
f
(
1
)
(
x
)
=
x
(
d
d
x
sin
x
)
+
(
d
d
x
x
)
sin
x
=
x
cos
x
+
sin
x
Similarity, for the second derivative
f
(
2
)
(
x
)
=
x
(
d
d
x
cos
x
)
+
(
d
d
x
x
)
sin
x
+
d
d
x
sin
x
=
−
x
sin
x
+
cos
x
+
cos
x
=
2
cos
x
−
x
sin
x
And further derivatives:
f
(
3
)
(
x
)
=
−
2
sin
x
−
(
x
cos
x
+
sin
x
)
=
−
3
sin
x
−
x
cos
x
f
(
4
)
(
x
)
=
−
3
cos
x
−
(
−
x
sin
x
+
cos
x
)
=
−
4
cos
x
+
x
sin
x
So, By exploiting the phase shift properties, we have:
y
(
n
)
=
n
sin
(
x
+
(
n
−
1
)
π
2
)
+
x
sin
(
x
+
n
π
2
)
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