यदि `y = | sinx |^(|x|)` है , तो `x = - (pi)/(6)` पर `(dy)/(dx)` का मान क्या है ?

Options

A. `(2^(-pi/6)(6 In 2 - sqrt(3)x))/(6)`
B. `(2^(pi/6) (6In 2 +sqrt(3)pi))/(6)`
C. `(2^(-pi/6)(6In 2 + sqrt(3)pi))/(6)`
D. `(2^(pi/6) (6 In 2 - sqrt(3)pi))/(6)`

Explanation

दिया है,
`y = |sin x|^(|x|)`
`:. X = - (pi)/(6), sin x lt 0, x lt 0`
`y = (-sinx)^(-x)`
`:' y = f(x)^(g(x))`
`:' (dy)/(dX) = f(x)^(0(x))`
`[(g(x))/(f(x)) f'(x) + log|f(x) |.g'(x)]`
` rArr (dy)/(dx) = (-sinx)^(-x)`
`[ ((-x))/((-sinx)). (-cosx) + log | (-sinx) |(-1)|]`
` rArr (dy)/(dx) |(x = - (pi)/(6)) = [-sin((-pi)/(6))]^(pi/6)`
`[((pi/6))/(-sin(-(pi)/(6))){-cos(-(pi)/(6))}-log|-sin(-(pi)/(6))]`
`= (sin'(pi)/(6))^(pi/6)[(-(pi)/(6))/(1/2)((sqrt(3))/(2)) - log (1/2)]`
` = (1/2)^(pi/6) [(-sqrt(3))/(6) - log'1/2]`
`= (2^(-(pi)/(6)))/(6) [ log_(2) - (sqrt(3)x)/(6)]`
`2 ^(-(pi)/(6))[(6log2-sqrt(3)pi)/(6)]`