Find `(dy)/(dx)a tx=-1,w h e n(sin"y")^(sin(pi/2x))+(sqrt(3))/2sec^(-1)(2x)+2^xtan(log(x+2))=0`

Options

A. `(3)/(pisqrt(pi^(2) - 3))`
B. `(-3)/(pisqrt(pi^(2) - 3))`
C. `(6)/(pisqrt(pi^(2) - 3))`
D. None of these

Answer

A

Explanation

Put `x = -1`
`(sin y)^(-1) + (sqrt(3))/(2) sec^(-1)sqrt(-2) + 2^(-1) xx 0 = 0`
`rArr (sqrt(3))/(2) (2x)/(3) = (sin y)^(-1)`
`sin y = -(sqrt(3))/(pi)`
Noq differentiate the equation
`y=|{:((siny)^(sin(pix)/(2))logsiny.cos'(pix)/(2).(pi)/(2)+(sqrt(3))/(2)(1)/(|2x|sqrt(4x^(2)-1))),(+2^(x)(sec^(2)log(x+2))/(x+2) + tanln(x+2)2^(x)ln2):}/((sinpix)/(2)(siny)^(sin'(pix)/(2)-1)cosy)|`
Put `x = -1` , `sin y = -sqrt(3) //pi, cos y = sqrt(1 - (3)/(pi^(2)))`
`(dy)/(dx) = -[(0+(sqrt(3))/(2sqrt(3))+1/2)/(-((-sqrt(3))/(pi))^(-2) (sqrt(pi^(2)-3))/(pi))]=(3)/(pisqrt(pi^(2)-3))`