Given y = ln((2x+3)/(7x^3 +1)). Find dy/dx

 y = ln(2x+3 / 7x^3 +1)

d/dx(2x+3 / 7x^3 + 1) by quotient rule which is(v.du/dx - u.dv/dx) / v^2  where u=2x+3 and v=7x^3 +1   gives (-27x^3 -63x^2 +2) / (7x^3 +1)^2

so d/dx(ln(2x+3 / 7x^3 +1) = ( (-27x^3 -63x^2 +2) / (7x^3 +1)^2 ) / (2x+3)/(7x^3 +1)

= ((-27x^3 -63x^2 +2) / (7x^3 +1) ) / (2x+3)

= (-27x^3 -63x^2 +2) / (7x^3 +1).(2x+3)

which is the final solution, since it cannot be simplified further.