Let `y` be an implicit function of `x` defined by `x^(2x)-2x^xcot y-1=0.` Then `y '(1)` equals: `1` b. `log2` c. `-log2` d. `-1`

Options

A. `-1`
B. `1`
C. `log2`
D. `-log2`

Answer

A

Explanation

`x^(2x)-2x^(x)coty-1=0`
At `x=1,1-2coty-1=0`
`impliescoty=0impliesy=(pi)/2`
Ondifferentiating Eq. (i) w.r.t `x` we get
`2x^(2x)(1+logx)-2[x^(x)(-cosec^(2)y)(dy)/(dx)+cot y x^(x)(1+log x )]=0`
At `(1,(pi)/2),2(1+log1)-2`
`[1(-1)((dy)/(dx))_((1,(pi)/2))+0]=0`
`implies 2+2 ((dy)/(dx))_((1,(pi)/2))=0`
`implies((dy)/(dx))_((1,(pi)/2))=-1`