text: Now the modular general case  is worked out. For this purpose, the derivatives of the complete elliptic integrals are derived after the modulus  and then they are combined. And then the Legendre's identity balance is determined.
  Because the derivative of the circle function is the negative product of the identical mapping function and the reciprocal of the circle function:
  These are the derivatives of K and E shown in this article in the sections above:
  In combination with the derivative of the circle function these derivatives are valid then:
  Legendre's identity includes products of any two complete elliptic integrals. For the derivation of the function side from the equation scale of Legendre's identity, the Product rule is now applied in the following:
  Of these three equations, adding the top two equations and subtracting the bottom equation gives this result:
  In relation to the  the equation balance constantly gives the value zero.
  The previously determined result shall be combined with the Legendre equation to the modulus  that is worked out in the section before:
  The combination of the last two formulas gives the following result:
  Because if the derivative of a continuous function constantly takes the value zero, then the concerned function is a constant function. This means that this function results in the same function value for each abscissa value  and the associated function graph is therefore a horizontal straight line.