g(x)=(√x)(sin(x)).
Let u=√x=x½ and v=sin(x), so du/dx=½x-½ and dv/dx=cos(x).
dg/dx=d(uv)/dx=vdu/dx+udv/dx (product rule)=(sin(x))(½/√x)+(√x)(cos(x)).
This can be written: dg/dx=sin(x)/(2√x)+cos(x)√x, or:
dg/dx=½sin(x)√x/x+cos(x)√x, or (½sin(x)/x+cos(x))√x.