Differentiate `(x sinx)^x` with respect to `xdot`

Answer

`(x cos x)^(x)[( log x +1) +{ log cos x- x cot x}]`

Explanation

Let `y = (x cos x)^(x).`
Taking logarithm on both sides, we get
`log y = log (x cos x)^(x)`
`=x log ( x cos x )`
`=x log x+ x log cos x`
Differentiating both sides with respect to x, we get
`(1)/(y)(dy)/(dx)=xcdot(1)/(x)+ log x+log cos x+xcdot ((-sin x)/(cos x))`
`" or "(dy)/(dx)=( x cos x)^(x)[(log x +1)+{ log cos x - x cot x}]`