Question

Find `(dy)/(dx)` at `x=-1` ,when `(siny)^(sin((pi/2)x))+sqrt3/2 sec^(-1)(2x)+2^x tan(ln(x+2))=0`

Answer 1

Correct Answer - `3/(pisqrt(pi^(2)-3)) `
Here, `(siny)^(sin.pi/2x) +sqrt3/2 sec^(-1)(2x)+2^(x) tan {log (x+2)}=0`
On differentiating both sides, we get
`(sin y)^(sin.pi/2 x)*log (sin y)*cos.pi/2 x*pi/2`
`" "+(sin. pi/2x)(sin y)^((sin .pi/2 x)-1)*cos y *(dy)/(dx) `
`" "+sqrt3/2*2/((2|x|)sqrt(4x^(2)-1))+(2^(x)*sec^(2){log (x+2)})/((x+2))`
`" "+ 2^(x) log 2* tan {log (x+2)}=0`
Putting `(x=- 1, y = - sqrt3/pi)`, we get
`(dy)/(dx)=(-sqrt3/pi)^(2)/sqrt(1-(sqrt(3)/pi)^(2))=3/(pisqrt(pi^(2)-3))`