th
Find the nth order
Derivative of x³ e^-2x.sinx
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1) Write
e
x
cos
(
x
)
=
e
x
R
e
(
e
i
x
)
=
R
e
(
e
(
1
+
i
)
x
)
.
2) Calculate
n
-th derivative of
e
(
1
+
i
)
x
:
d
n
d
x
n
e
(
1
+
i
)
x
=
(
1
+
i
)
n
e
(
1
+
i
)
x
.
3) Take the real part :
d
n
d
x
n
e
x
cos
(
x
)
=
R
e
(
(
1
+
i
)
n
e
(
1
+
i
)
x
)
=
e
x
R
e
(
(
1
+
i
)
n
e
i
x
)
.
To simplify that, you have to write
(
1
+
i
)
=
√
2
e
i
π
4
(trigonometric form of
1
+
i
). So,
(
1
+
i
)
n
e
i
x
=
√
2
n
e
i
(
n
π
4
+
x
)
=
2
n
2
(
cos
(
n
π
4
+
x
)
+
i
sin
(
n
π
4
+
x
)
)
Finally, taking real part,
d
n
d
x
n
e
x
cos
(
x
)
=
2
n
2
e
x
cos
(
n
π
4
+
x
)
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