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Byju's Answer
Standard XII
Mathematics
How to Find the Inverse of a Function
State whether...
Question
State whether the statement is true/false.
If
y
=
x
x
, then
d
2
y
d
x
2
−
1
y
(
d
y
d
x
)
2
y
x
=
1
A
True
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B
False
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Solution
The correct option is
B
False
y
=
x
x
Taking log on both sides,
log
y
=
x
log
x
Differentiate w.r.t
x
,
we have
1
y
(
d
y
d
x
)
=
x
×
1
x
+
log
x
d
y
d
x
=
y
+
y
log
x
d
y
d
x
=
y
(
1
+
log
x
)
----- ( 1 )
Again differentiating w.r.t.
x
,
⇒
d
2
y
d
x
2
=
y
×
1
x
+
(
1
+
log
x
)
d
y
d
x
⇒
d
2
y
d
x
2
=
y
x
+
y
(
1
+
log
x
)
2
----- ( 2 )
We have given,
d
2
y
d
x
2
−
1
y
(
d
y
d
x
)
2
y
x
=
1
L.H.S
=
d
2
y
d
x
2
−
1
y
(
d
y
d
x
)
2
y
x
=
y
x
+
y
(
1
+
log
x
)
2
−
1
y
×
y
2
(
1
+
log
x
)
2
−
y
x
[ From ( 1 ) and ( 2 ) ]
=
0
∴
L
.
H
.
S
≠
R
.
H
.
S
∴
d
2
y
d
x
2
−
1
y
(
d
y
d
x
)
2
y
x
=
1
is false.
Suggest Corrections
0
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