`(logx)^(x)+x^(logx)`

Explanation

`y=x^(logx)+(logx)^(x)" (माना )"`
`=u+v`
`rArr" "(dy)/(dx)=(du)/(dx)+(dv)/(dx)" …(1)"`
अब `u=x^(logx)`
`rArr" "logu=log(x^(logx))=logx.logx=(logx)^(2)`
`rArr" "(1)/(u)(du)/(dx)=(2logx)/(x)`
`rArr" "(du)/(dx)=(2ulogx)/(x)=(2logx)/(x).x^(logx)`
और `v=(logx)^(x)`
`rArr" "logv=log(logx)^(x)=xlog(logx)`
`rArr" "(1)/(v)(dv)/(dx)=(x)/(xlogx)+log(logx)`
`rArr" "(dv)/(x)=v[(1)/(logx)+log(logx)]`
`=(logx)^(x)[(1)/(logx)+log(logx)]`
समीकरण (1 ) से
`(dy)/(dx)=x^(logx).(2logx)+(logx)^(x)[(1)/(logx)+log(logx)].`