`(sinx)^(x)+sin^(-1)sqrtx`

1.	`(sinx)^(x)+sin^(-1)sqrtx`
Answer» माना `y=(sinx)^(x)+sin^(-1)sqrtx` माना `u=(sinx)^(x)` तथा `v=sin^(-1)sqrtx` `therefore" "y=u+v` `rArr" "(dy)/(dx)=(du)/(dx)+(dv)/(dx)" …(1)"` अब `u(sinx)^(x)` `rArr" "logu=log(sinx)^(x)=xlog sin x` `rArr" "(1)/(u)(du)/(dx)=x(d)/(dx)logsinx+log sin x(d)/(dx)x` `rArr" "(du)/(dx)=u[x.(cosx)/(sinx)+log sinx.1]` `rArr" "(du)/(dx)=(sinx)^(x)[x cot x+log sin x]` तथा `v=sin^(-1)sqrtx` `rArr" "(dv)/(dx)=(d)/(dx)sin^(-1)sqrtx` `=(1)/(sqrt(1-(sqrtx))^(2))(d)/(dx)sqrtx=(1)/(2sqrtx(1-x))` समीकरण (1 ) से `(dy)/(dx)=(sinx)^(x)[x cot x+log sin x]+(1)/(2sqrtx(1-x))`

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