निम्न फलनों का x के सापेक्ष अवकल गुणांक ज्ञात कीजिए|
` (i) tan ^(-1) (1+ x+x^(2))`
` (ii) )(sin ^(-1) x)^(2) -(cos ^(-1) x)^(2)`
` (iii) llog tan ^(-1) x^2)`

Explanation

माना ` y= tan ^(-1)(1+ x+x^(2) ),` माना ` (1+ x+x^(2) ) =t`
` therefore y= tan ^(-1) t `
` therefore (dy)/(dx) =(d)/(dt ) tan ^(-1) t (d)/(dx)(1+ x+x^(2)) `
` " "= (1)/(1+ t^(2) )(0 +1 +2x) ` ltbr gt` " "= ( 1+ 2x) /( 1+ (1+ x+x^(2)) ^(2))`
माना `y=(sin ^(-1) x)^(2) -(cos ^(-1) x)^(2) `
माना `sin ^(-1) x=t_1` तथा `cos ^(-1) x=t_2 `
` therefore " " y= t _1^(2) -t_2^(2) `
` therefore (dy)/(dx)=(d)/(dt_1 )t_1^(2) (d)/(dx) t_1 -( d)/(dt_2 ) t_2^(2) (d)/(dx) t_2`
` " "=2t _1 (d)/(dx) sin ^(-1) x- 2t_2 (d)/(dx) cos ^(-1) x `
` = (2sin ^(-1) x )/( sqrt ( 1-x^(2)) ) + (2cos ^(_1) x)/( sqrt (1-x^(2)))`
` " "= (2) /(sqrt ( (1-x^(2)) ) )[ sin ^(-1) x + cos ^(-1) x ] `
(iii) माना ` y= log tan ^(-1) x^(2) , `माना ` tan ^(-1) x^(2) =t `
` therefore y= log t `
` therefore *(dy)/(dx)= (d)/(dt )log t (dt)/(dx) =(1)/(t) (d)/(dx) tan ^(-1) x^(2) `
` =(1)/(t) *( 1) /(1+(x^(2) )^(2) )*2x =(2x) /((1+ x^(4))tan ^(-1) x^(2))`