Question 15
Last updated at Dec. 21, 2017 by Teachoo

If y = log (√x + 1 / √x)
^{
2
}
then prove that x (x + 1)
^{
2
}
y
_{
2
}
+ (x + 1)
^{
2
}
y
_{
1
}
= 2
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Transcript

Question 15
If y = log (โ๐ฅ+1/โ๐ฅ)^2then prove that x (x + 1)2 y2 + (x + 1)2 y1 = 2
Given
y = log (โ๐ฅ+1/โ๐ฅ)^2
Using log a2 = 2 log a
y = 2 log (โ๐ฅ+1/โ๐ฅ)
y = 2 log ((โ๐ฅ ร โ๐ฅ + 1)/โ๐ฅ)
y = 2 log ((๐ฅ + 1)/โ๐ฅ)
Using log ๐/๐ = log a โ log b
y = 2 log (๐ฅ+1) โ 2 log โ๐ฅ
y = 2 log (๐ฅ+1) โ 2 log (๐ฅ)^(1/2)
Using log a2 = 2 log a
y = 2 log (๐ฅ+1) โ 2 ร 1/2 log ๐ฅ
y = 2 log (๐ฅ+1) โ log ๐ฅ
Now finding y1 and y2
From (1)
y = 2 log (๐ฅ+1) โ log ๐ฅ
Differentiating w.r.t x
๐๐ฆ/๐๐ฅ = 2 ๐(logโก(๐ฅ + 1) )/๐๐ฅ โ ๐(logโก๐ฅ )/๐๐ฅ
๐๐ฆ/๐๐ฅ = 2 1/(๐ฅ + 1) โ 1/๐ฅ
๐๐ฆ/๐๐ฅ = 2/(๐ฅ + 1) โ 1/๐ฅ
๐๐ฆ/๐๐ฅ = (2๐ฅ โ(๐ฅ +1))/(๐ฅ(๐ฅ + 1))
๐๐ฆ/๐๐ฅ = ((๐ฅ โ 1))/(๐ฅ(๐ฅ + 1))
y1 = ((๐ฅ โ 1))/(๐ฅ(๐ฅ + 1))
Now, from (2)
y1 = ((๐ฅ โ 1))/(๐ฅ(๐ฅ + 1))
Differentiating w.r.t x
y2 = (((๐ฅ โ 1))/๐ฅ(๐ฅ + 1) )^โฒ
= x (x + 1)2 ร (ใโ๐ฅใ^2 +2๐ฅ + 1)/(๐ฅ^2 (๐ฅ + 1)^2 ) + (x + 1)2 ร ((๐ฅ โ 1))/(๐ฅ(๐ฅ + 1))
= (ใโ๐ฅใ^2 +2๐ฅ + 1)/๐ฅ + (x + 1) ร ((๐ฅ โ 1))/๐ฅ
= (ใโ๐ฅใ^2 +2๐ฅ + 1)/๐ฅ + ((๐ฅ + 1)(๐ฅ โ 1))/๐ฅ
= (ใโ๐ฅใ^2 +2๐ฅ + 1)/๐ฅ + (๐ฅ^2 โ 1)/๐ฅ
= (ใโ๐ฅใ^2 +2๐ฅ + 1 + ๐ฅ^2 โ 1)/๐ฅ
= 2๐ฅ/๐ฅ
= 2
= RHS
Hence proved

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