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the value of ddxx2x
Question:
The value of
d
d
x
x
2
x
=
MHT CET
Updated On:
Aug 14, 2024
(A)
2
x
2
log
x
(B)
x
2
2
(
1
+
log
x
)
(C)
x
2
x
(D)
2
x
2
x
[
1
+
log
x
]
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The Correct Option is
D
Solution and Explanation
Explanation:
Given:
d
d
x
x
2
x
Concept:
log
x
n
=
n
log
x
d
d
x
[
log
x
]
=
1
x
Let,
y
=
x
2
x
Taking log on both the sides, we get
log
y
=
log
(
x
2
x
)
=
2
x
log
x
(
∵
log
x
n
=
n
l
o
g
x
)
Now, taking derivatives,
d
d
x
[
log
y
]
=
2
{
d
d
x
[
x
]
log
x
+
d
d
x
[
log
x
]
x
}
1
y
d
y
d
x
=
2
[
log
x
+
x
⋅
1
x
]
(
∵
d
d
x
[
log
x
]
=
1
x
)
⇒
d
y
d
x
=
2
y
[
1
+
log
x
]
⇒
d
y
d
x
=
2
x
2
x
[
1
+
log
x
]
(
∵
y
=
x
2
x
)
Hence, the correct option is (D).
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