1.

Differentiate the following w.r.t. x:`(logx)^x+x^(logx)`

Answer» `y = x^("log" x) + ("log" x)^(x)`
`=u + v " " (say)`
`implies (dy)/(dx) = (du)/(dx) + (dv)/(dx) " " ….(1)`
`"Now",u=x^("log"x)`
`rArr " log"u-"log"(x^("log"x))-"log"x*"log"x-("log"x)^(2)`
`rArr (1)/(u)(du)/(dx) = (2"log"x)/(x)`
`rArr (du)/(dx) = (2u "log"x)/(x) = (2"log"x)/(x).x^("log"x)`
`"and "v = ("log"x)^(x)`
`rArr "log"v = "log" ("log"x)^(x) = x"log" ("log"x)`
`rArr (1)/(v) (dv)/(dx) = (x)/(x"log"x) + "log"("log"x)`
`rArr (dv)/(dx) = v[(1)/("log"x) + "log"("log"x)]`
`=("log"x)^(x) [(1)/("log"x) + "log"("log"x)]`
From equation (1)
`(dy)/(dx) = x^("log"x)*(2"log"x)/(x) + ("log"x)^(x)[(1)/("log"x) + "log"("log"x)]`


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