Differentiate ` x^(x) + (sin x)^(logx) ` w.r.t. x ,` x gt 0 `

Explanation

` y = x^(x) + (sin x ) ^(log x)`
` (dy)/(dx) = d/ (dx) (x^(x)) + d/(dx) ((sin x) log x)`
` u= x^(x) and v = (sin x)^(log x)`
log u= x log x and log v = ( log x ) log ( sin x)
` 1/u (du)/(dx) = (x/x + log x) , 1/v (dv)/(dx) = (log (sin x))/ x + ((log x) (cos x))/ (sin x ) `
` (du)/(dx) = x^(x) (1 + log x) , (dv)/(dx) = ( sin x ) ^(log x) (( log(sin x))/ x + log x cot x)` ltbrrgt ` (dy)/(dx) = x^(x) (1 + log x) + (sin x) ^(log x) ( (log sin x)/x + log x cot x)`