if `y=log_(sinx) tanx` then `((dy)/(dx)) _(pi/4)` is

Options

A. `(4)/(log 2)`
B. `-(4)/(log 2)`
C. ` (1)/(log 2)`
D. None of these

Answer

B

Explanation

We have , ` y = log _(sin x) (tan x ) = (" log tan x")/("log sin x")`
`therefore dy/dx = (log sinx . (1)/(tanx). sec^(2) x - log tan x . (1)/(sin x) cos x )/(log sin x)^(2)`
At `x = (pi)/(4) , ((dy)/(dx))_(x = (pi)/(4)) = (log ((1)/(sqrt(2)))(sqrt(2))^(2))/((log (1)/(sqrt(2)))^(2)) = (sqrt(2))^(2)/(log ((1)/(sqrt(2))))`
` = - (2xx2)/(log 2) = - (4)/(log 2)`