h'(x)=x^4/x+4x^3ln(x)=x^3+4x^3ln(x)
h''(x)=3x^2+4x^3/x+12x^2ln(x)=7x^2+12x^2ln(x)
h'''(x)=14x+24xln(x)
h''''(x)=14+24ln(x)+24=38+24ln(x)
The 5th derivative is simply 24/x, and thereafter -24/x^2, 48/x^3, -144/x^4, 576/x^5 and so on, wih alternating positive and negative and the coefficient building: 24, 2*24, 2*3*24, 4!*24. The nth derivative is (-1)^(n+1)(n-5)!24/x^(n-4), where n>4.