Find `(dy)/(dx)`, when:
`y=(xcosx)^(x)+(x sin x)^(1//x)`

Answer

`(x cos x)^(x){1-x tan x+log x+log cos x}+(x sin x)^(1//x).{(1)/(x^(2))+(cotx)/(x)-(logx)/(x^(2))-(logsinx)/(x^(2))}`

Explanation

Let `y=u+v`, where `u=(xcosx)^(x) and v=(x sin x)^(1//x).`
Then, `(dy)/(dx)=(du)/(dx)+(dv)/(dx)`.
Now, `u=(x cos x)^(x) rArr log u=x log (x cos x)`
`rArr logu=x.{logx+log cos x}`
`rArr(1)/(u).(du)/(dx)=x.{(1)/(x)-(sinx)/(cosx)}+{logx+log cos x}.1`
`rArr (du)/(dx)=u{(1-x tanx)+(log x+log cosx)}`
`rArr (du)/(dx)=(x cos x)^(x){(1-x tan x+logx+log cos x)}.`
And, `v=(x sin x)^(1//x) rArr log v=(1)/(x)log(x sinx)`
`rArr logv=(1)/(x).{log x+log sin x}`
`rArr (1)/(v).(dv)/(dx)=(1)/(x).((1)/(x)+(cos x)/(sinx))+(logx+log sin x)((-1)/(x^(2)))`
`rArr(1)/(v).(dv)/(dx)={(1)/(x^(2))+(1)/()cotx-(logx)/(x^(2))-(logsinx)/(x^(2))}`
`rArr(dv)/(dx)=v.{(1)/(x^(2))+(cotx)/(x)-(logx)/(x^(2))-(log sin x)/(x^(2))}`
`rArr (dv)/(dx)=(x sin x)^(1//x).{(1)/(x^(2))+(cotx)/(x)-(logx)/(x^(2))-(log sin x)/(x^(2))}.`