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If u, v be implicit functions of x, y such that fi(u,v.x,y) = 0,f(u, v, x,y) = 0
is given by
Answers
Answer
Correct option is
A
−1
Now x=1⇒1−2coty−1=0
⇒coty=0 or y=
2
π
Let u,v,w be the functions of x then ⇒u+v+w=0 and
dx
du
+
dx
dv
+
dx
dw
=0
Take u=x
2x
⇒logu=logx
2x
=2xlogx using the power form of logarithm
⇒
u
1
dx
du
=2[x×
x
1
+logx]=2(1+logx)
∴
dx
du
=2u(1+logx)=2x
2x
(1+logx)
Take v=−2x
x
coty
dx
dv
=−2[x
2
(−csc
2
y)
dx
dy
+coty
dx
d(x
x
)
]
=−2[x
2
(−csc
2
y)
dx
dy
+x
x
coty(1+logx)]
Take w=−1
dx
dw
=
dx
d(−1)
=0 since differential of a constant=0
Now,
dx
du
+
dx
dv
+
dx
dw
=0
⇒2x
2x
(1+logx)−2[x
2
(−csc
2
y)
dx
dy
+x
x
coty(1+logx)]+0=0
⇒
dx
dy
=
−2x
2
csc
2
y
2x
2x
(1+logx)−2x
2
(1+logx)coty
⇒[
dx
dy
]
(1,
2
π
)
=
−2×1
2
csc
2
2
π
2×1
2
(1+log1)−2×1
2
(1+log1)cot
2
π
⇒y
′
(1)=
−2
1(1+log1)
sin
2
2
π
since
csc
2
2
π
1
=sin
2
2
π
∴y
′
(1)=−1×1=−1 since sin
2
π
=1