यदि `y=(xsin^(-1)x)/(sqrt(1-x^(2)))+logsqrt(1-x^(2))`, तब सिद्ध कीजिए कि- `(dy)/(dx)=(sin^(-1)x)/((1-x^(2))^(3//2))`

Explanation

दिया गया है-
`y=(xsin^(-1)x)/(sqrt(1-x^(2)))+logsqrt(1-x^(2))`
`rArry=(xsin^(-1)x)/(sqrt(1-x^(2)))+1/2log(1-x^(2))`
दोनों पक्षों का x के सापेक्ष अवकलन करने पर,
`(dy)/(dx)=((sqrt(1-x^(2)))d/(dx)(xsin^(-1)x)-(xsin^(-1)x)d/(dx)(sqrt(1-x^(2))))/((1-x^(2)))+1/2d/(dx)log(1-x^(2))`
`rArr(dy)/(dx)=((sqrt(1-x^(2))).{x.1/(sqrt(1-x^(2)))+sin^(-1)x.1}-(xsin^(-1)x).1/(2sqrt(1-x^(2)))(-2x))/((1-x^(2)))+1/2(.1)/((1-x^(2)))(-2x)`
`rArr(dy)/(dx)=((x+sqrt(1-x^(2))sin^(-1)x)+(x^(2)sin^(-1)x)/(sqrt(1-x^(2))))/((1-x^(2)))-x/((1-x^(2)))`
`rArr(dy)/(dx)=(xsqrt(1-x^(2))+(1-x^(2))sin^(-1)x+x^(2)sin^(-1)x)/((1-x^(2))sqrt(1-x^(2)))-x/((1-x^(2)))`
`rArr(dy)/(dx)=(xsqrt(1-x^(2))+sin^(-1)x)/((1-x^(2))sqrt(1-x^(2)))-x/((1-x^(2)))`
`rArr(dy)/(dx)=(xsqrt(1-x^(2))+sin^(-1)x-xsqrt(1-x^(2)))/((1-x^(2))sqrt(1-x^(2)))` ltbr gt`rArr(dy)/(dx)=(sin^(-1)x)/((1-x^(2))^(3//2))` यही सिद्ध करना था|