निम्न फलनों का x के सापेक्ष अवकलन कीजिए|
` (i) y= tan ^(-1) ((3a^(2)x-x^(3))/(a(a^(2)-3x^(2))))`
` (ii) y=sin ^(-1) [x sqrt(1-x)-sqrtxsqrt((1-x^(2)))`

Explanation

` (i) y= tan ^(-1) [(3a^(2) x-x^(3) )/(a(a^(2) -3x^(2))]]`
माना ` x=a tan theta " "rArr theta =tan ^(-1) ((x)/(a))`
` therefore " "y=tan ^(-1) [(a^(3) (3tan theta -tan ^(3) theta ))/(a^(3) (1-3 tan ^(2) theta ))]`
` =tan ^(-1) tan 3 theta =3 theta `
`rArr y= 3 tan ^(-1) ((x)/(a)) `
` therefore (dy)/(dx) =3(1)/(1+((x)/(a))^(2))*(1)/(a) rArr (dy)/(dx) =(3a)/(a^(2)+x^(2))`
`" " y=sin ^(1) [x sqrt (1-x) -sqrt( x)sqrt (1-x^(2))`
माना ` x= sin alpha rArr alpha = sin ^(-1) x `
तथा ` sqrt x =sin beta " "rArr beta =sin ^(-1) sqrt(x) `
` therefore y= sin ^(-1) [sin alpha sqrt(1-sin ^(2) beta ) -sin beta sqrt (1-sin ^(2) alpha ]`
`" "= sin ^(-1) [sin alpha cos beta - cos alpha sin beta ]`
` " "= sin ^(-1) sin (alpha -beta )= alpha - beta `
` " "y= sin ^(-1) x-sin ^(-1) sqrt(x)`
` therefore (dy)/(dx) =(d)/(dx)sin ^(-1) x-(d)/(dx) sin ^(-1) (x)^(1//2) `
` (dy)/(dx) =(1) /sqrt(1-x^(2))-(1)/(2sqrt xsqrt( 1-x) )`