निम्न फलन का x के सापेक्ष अवकलन कीजिए|
` (i) y= tan ^(-1) ((sqrt(1+x^(2) -1))/x)`
` (ii) y=sin [2tan ^(-1) sqrt(((1-x)) /((1+x))`

Explanation

(i ) ` y= tan ^(-1) [(sqrt(1+x^(2) )-1)/(x) ]`
माना ` x= tan theta rArr theta tan ^(-1) x `
` therefore " "y= tan ^(-1) [(sqrt((1+ tan ^(2) theta ))-1)/(tan theta )] =tan ^(-1) [(sectheta -1)/(tan theta )]`
` rArr " "y= tan ^(-1) [(1-cos theta )/(sin theta ) ] =tan ^(-1) tan((theta )/(2))=(theta )/(2)`
` rArr " "y= (1)/(2) tan ^(-1) x `
` rArr " "(dy)/(dx) =(1)/(2) (d)/(dx) tan ^(-1) x=(1)/(2) ((1)/(a+x^(2)))`
` therefore (dy)/(dx) =(1)/(2(1+x^(2)))`
(ii) ` " "y= sin [2tan ^(-1) sqrt(((1-x))/((1+x)))]`
माना ` x= cos theta rArr theta cos ^(-1) x `
` therefore " "=sin [2tan ^(-1)sqrt(((1-cos theta )/(1+costheta )))] `
` =sin [ 2tan ^(-1) sqrt(((1-1+2sin ^(2)""(theta)/(2))/(1+2cos ^(2) ""(theta )/(2) -1)))]`
` =sin [2tan ^(-1) (tan ""(theta )/(2))] =sin theta `
` rArr y=sin theta =sin (cos ^(-1) x )`
` therefore (dy)/(dx) =(d)/(dx) sin (cos ^(-1) x ) =cos (cos ^(-1) x ) ((-1)/(sqrt(1-x^(2))))`
` therefore (dy)/(dx) =(-x) /(sqrt((1-x^(2) ) ))`