`x^(sin^(-1)x)` का अवकलन `sin^(-1)x` के सापेक्ष कीजिए

Explanation

माना `u=x^(sin^(-1)x)`
`rArrlogu=sin^(-1)xlogx`
[लघुगणक लेने पर]
`therefore 1/u(du)/(dx)=sin^(-1)x1/x+logx.(1)/(sqrt(1-x^(2)))`
`rArr(du)/(dx)=u[(sin^(-1)x)/x+(logx)/(sqrt(1-x^(2))]`
`rArr(du)/(dx)=u[(sin^(-1)x)/x+(logx)/(sqrt(1-x^(2)))]`
`rArr(du)/(dx)=x^(sin^(-1)x)[(sin^(-1)x)/x+(logx)/(sqrt(1-x^(2)))]`
और `v=sin^(-1)x`
`therefore(dv)/(dx)=1/(sqrt(1-x^(2)))`
अब, `(du)/(dx)=(du//dx)/(dv//dx)`
`rArr(du)/(dx)=(x^(sin^(-1)x)[(sinx)/x+(logx)/(sqrt(1-x^(2)))])/(1/(sqrt(1-x^(2))))`
`rArr(du)/(dx)=x^(sin^(-1)x)[(sqrt(1-x^(2)))/xsinx+logx]`