यदि ` y= sin ^(-1) ((6x -4sqrt (1-4x^(2)))/(5))` तब ` (dy)/(dx)` का मान ज्ञात कीजिए|

Explanation

दिया है- `y=sin ^(-1) ((6x -4 sqrt (1-4x^(2)))/(5))`

अब, `x=(1)/(2)sin theta ` रखने पर

`y=sin ^(-1) [(6xx(sin theta //2) -4sqrt (1-4xx(sin theta//2)))/(5)]`

` =sin ^(-1) ((3sintheta -4sqrt (1-sin ^(2)theta ))/(5))`

` =sin ^(-1) ((3sin theta -4cos theta )/(5))`

` =sin ^(-1) ((3 )/(5)sin theta -(4)/(5) cos theta )" "...(1)`

माना ` cos phi =(3)/(5) ,` तब
` sin phi =sqrt (1-cos ^(2)phi )`

`=sqrt(1-((3)/(2) )^(2))`
` =sqrt (1-(9)/(25))`

`= sqrt((16)/(25))= (4)/(5)`

समीकरण `(1 )` से,

` y= sin ^(-1) (cos phi sin theta -sin phi cos theta ) `

` =sin ^(-1) [sin (theta -phi ) ] ``=theta -phi `

` rArr y=sin ^(-1) (2x) -cos ^(-1) ((3)/(5))`

` [`चूंकि ` x= (1)/(2) sin theta `
`rArr sin theta =2x`

` rArr theta =sin ^(-1) (2x) `
तथा `cos phi =(3)/(5) rArr phi =cos^(-1)"" (3)/(5)]`

`x` के सापेक्ष अवकलन करने पर

` (dy)/(dx) =(1)/(sqrt((1-2x)^(2)))(d)/(dx) (2x) -0`

` =(2)/(sqrt(1-4x^(2)))`