` {x^(tan x )+sqrt((x^(2) +1)/(x) )}` को `'x'` के सापेक्ष अवकलित कीजिए|

Explanation

माना ` " "y_1 =x^(tan x,) " "y_2 =sqrt((x^(2) +1)/(x))`
तथा ` " "y= y_1 +y_2`
दोनों पक्षों का log लेने पर,
` rArr " "log y_1 =(tan x )(log x ) `
अब ,x के सापेक्ष अवकलित करने पर
` rArr (1)/(y_1) (dy_1)/(dx) =(tan x )*(d)/(dx) (log x )+ (log x) *(d)/(dx) (tan x ) `
` " "=(tan x )* (1)/(2) +(logx )sec^(2) x `
` rArr " "(dy_1)/(dx) =y_1 [(tanx )/(x)+ (log x) sec ^(2) x ]`
` " "=x tan x {(tan x)/(x) +(log x ) sec^(2) x}`
अतः ` " "y^(2) =sqrt((x^(2) +1) /(x))`
दोनों पक्षों का log लेने पर,
` log y_2=(1)/(2) {log (x^(2)+1) -log x}`
x के सापेक्ष अवकलन करने पर,
` (1)/(y_2) (dy_2)/(dx)=(1)/(2) {(2x)/((x^(2)+1))-(1)/(x)}`
` (dy_2)/(dx) =(y_2)/(2) {(2x^(2)-(x^(2)+1) )/(x(x^(2) +1))}`
` =(1)/(2) sqrt((x^(2)+1) /(x)) {((x^(2) -1)) /(x(x^(2)+1))}`
परन्तु `y=y_1 +y_2 `
` 'x'` के सापेक्ष अवकलन करने पर
` rArr (dy)/(dx) =(dy_1)/(dx) +(dy_2)/(dx)`
` =x^(tan x ) *{(tanx )/(x)+(logx ) sec^(2)x }`
` + (1)/(2) sqrt((x^(2)+1)/(x))*{((x^(2)-1))/(x(x^(2)+1))}`