Let u=x3-7x2 and v=√(1+9x2), then:
du/dx=3x2-14x and dv/dx=9x/√(1+9x2).
dy/dx=v(du/dx)+u(dv/dx)=(3x2-14x)√(1+9x2)+9x(x3-7x2)/√(1+9x2).
This can be rationalised partly:
(3x2-14x)√(1+9x2)+9x(x3-7x2)√(1+9x2)/(1+9x2),
and then simplified:
[(3x2-14x)(1+9x2)+9x(x3-7x2)]√(1+9x2)/(1+9x2),
(3x2+27x4-14x-126x3+9x4-63x3)√(1+9x2)/(1+9x2),
x(36x3-189x2+3x-14)√(1+9x2)/(1+9x2).