Find the roots of the quadratic equation by differentiation method

Can you elaborate what you mean? Your post is not actually clear. Different from what? From Quadratic formula? There are many ways to find the roots of a quadratic equation.

  1. By calculator using Shift Solve
  2. Using the quadratic function of calculator
  3. By factoring
  4. By factor theorem
  5. By quadratic formula
  6. By inspection

About roots of equations, quadratic or non-quadratic, advance methods are available. Example is Newton's method and my personal favorite is bisection method. Are you looking for more advanced method?