`(log x)^(tan x ) ` का ` sin (mcos ^(-1) x ) `के सापेक्ष अवकल गुणांक ज्ञात कीजिए

Explanation

माना ` y_1 =(log x ) ^(tan x ) ` तथा ` y_2 sin (mcos ^(-1) x )`
` because " "y_1 =(log x )_^(tan x ),` दोनों पक्षों का लघुगणक लेने पर,
` log y_1 =tan x log x `
` therefore " "(1)/(y) (dy)/(dx) =(tan x )/(xlog x ) +sec ^(2) x log log x `
` rArr " "(dy_1) /(dx) =(log x )^(tan x ) [(tan x )/(xlog x ) +sec ^(2) x log log x ]`
तथा ` " "y_2 =sin (m*cos ^(-1) x ) `
` therefore " "(dy_2)/(dx) =(d)/(dx) sin (mcos ^(-1) x )`
या ` (dy_2)/(sx) =(-mcos (mcos ^(-1) x ))/(sqrt((1-x^(2))))`
अतः` (dy_1)/(dy_2) =(y_1 ("का x के सापेक्ष अवकलन")) / (y_2("का x के सापेक्ष अवकलन"))`
` =((logx )^(tan x) [(tan x)/(xlog x ) +sec ^(2) xlog log x ])/(-(mcos (mcos ^(-1)x))/(sqrt(1-x^(2))))`
` =- ((log x)^(tan x) [(tan x)/(xlog x )+sec ^(2) x log log x ]sqrt(1-x^2))/(mcos (mcos ^(-1)x))`