f(x)=√x^(√x)-xln(1-3x), df/dx needed.
Let v=√x^(√x)=x^(½√x)=e^(½√xln(x)).
Let u=½√xln(x), du/dx=½√x/x+ln(x)/(4√x); v=e^u, dv/du=e^u.
dv/dx=(dv/du)(du/dx)=e^u(½√x/x+ln(x)/(4√x)),
dv/dx=√x^(√x)(½√x/x+ln(x)/(4√x))=¼√x/x(2+ln(x))√x^(√x).
df/dx=dv/dx+3x/(1-3x)-ln(1-3x),
df/dx=¼√x/x(2+ln(x))√x^(√x)+3x/(1-3x)-ln(1-3x).