By definition the derivative of f(x)f(x) is:
lim_(Deltax->0) (f(x+Deltax)-f(x))/(Deltax) = lim_(Deltax->0) (Deltaf)/(Deltax)
Calculate:
Deltaf = (2-(x+Deltax))/(3(x+Deltax)+1) - (2-x)/(3x+1)
Deltaf =( (2-x-Deltax)(3x+1) -(3x+3Deltax+1) (2-x))/((3x+3Deltax+1)(3x+1))
Deltaf =( cancel((2-x)(3x+1))-Deltax(3x+1) -cancel((3x+1) (2-x))-3Deltax(2-x))/((3x+3Deltax+1)(3x+1))
Deltaf = ( -Deltax(3x+1+6-3x))/((3x+3Deltax+1)(3x+1)) = -(7Deltax)/((3x+Deltax+1)(3x+1))
Divide by Deltax:
(Deltaf)/(Deltax) = -7/((3x+Deltax+1)(3x+1))
Passing to the limit:
lim_(Deltax->0) (Deltaf)/(Deltax) = lim_(Deltax->0) -7/((3x+Deltax+1)(3x+1)) = - 7/((3x+1)^2)