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Differentiate x^(sin^-1x) with respect t...

Differentiate `x^(sin^-1x)` with respect to `x` :

Text Solution

Verified by Experts

Let `y=x^(sin^-1x)" …(i)"`
Taking logarithm on both sides of (i), we get
`log y=(sin^(-1)x)(logx)." …(ii)"`
On differentiating both sides of (ii) w.r.t. x, we get
`(1)/(y).(dy)/(dx)=(sin^(-1)x).(d)/(dx)(logx)+(logx).(d)/(dx)(sin^(-1)x)`
`=(sin^(-1)x)(1)/(x)+(logx).(1)/(sqrt(1-x)^(2))`
`rArr" "(dy)/(dx)=y.[(sin^(-1)x)/(x)+(logx)/(sqrt(1-x^(2)))]`
`rArr" "(dy)/(dx)=x^(sin^(-1)x).{(sin^(-1)x)/(x)+(logx)/(sqrt(1-x^(2)))}.`
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