code: def is_deriv_k(fa, fd, DE):
      r"""
      Checks if Df/f is the derivative of an element of k(t).
  
      Explanation
      ===========
  
      a in k(t) is the derivative of an element of k(t) if there exists b in k(t)
      such that a = Db.  Either returns (ans, u), such that Df/f == Du, or None,
      which means that Df/f is not the derivative of an element of k(t).  ans is
      a list of tuples such that Add(*[i*j for i, j in ans]) == u.  This is useful
      for seeing exactly which elements of k(t) produce u.
  
      This function uses the structure theorem approach, which says that for any
      f in K, Df/f is the derivative of a element of K if and only if there are ri
      in QQ such that::
  
              ---               ---       Dt
              \    r  * Dt   +  \    r  *   i      Df
              /     i     i     /     i   ---   =  --.
              ---               ---        t        f
           i in L            i in E         i
                 K/C(x)            K/C(x)
  
  
      Where C = Const(K), L_K/C(x) = { i in {1, ..., n} such that t_i is
      transcendental over C(x)(t_1, ..., t_i-1) and Dt_i = Da_i/a_i, for some a_i
      in C(x)(t_1, ..., t_i-1)* } (i.e., the set of all indices of logarithmic
      monomials of K over C(x)), and E_K/C(x) = { i in {1, ..., n} such that t_i
      is transcendental over C(x)(t_1, ..., t_i-1) and Dt_i/t_i = Da_i, for some
      a_i in C(x)(t_1, ..., t_i-1) } (i.e., the set of all indices of
      hyperexponential monomials of K over C(x)).  If K is an elementary extension
      over C(x), then the cardinality of L_K/C(x) U E_K/C(x) is exactly the
      transcendence degree of K over C(x).  Furthermore, because Const_D(K) ==
      Const_D(C(x)) == C, deg(Dt_i) == 1 when t_i is in E_K/C(x) and
      deg(Dt_i) == 0 when t_i is in L_K/C(x), implying in particular that E_K/C(x)
      and L_K/C(x) are disjoint.
  
      The sets L_K/C(x) and E_K/C(x) must, by their nature, be computed
      recursively using this same function.  Therefore, it is required to pass
      them as indices to D (or T).  E_args are the arguments of the
      hyperexponentials indexed by E_K (i.e., if i is in E_K, then T[i] ==
      exp(E_args[i])).  This is needed to compute the final answer u such that
      Df/f == Du.
  
      log(f) will be the same as u up to a additive constant.  This is because
      they will both behave the same as monomials. For example, both log(x) and
      log(2*x) == log(x) + log(2) satisfy Dt == 1/x, because log(2) is constant.
      Therefore, the term const is returned.  const is such that
      log(const) + f == u.  This is calculated by dividing the arguments of one
      logarithm from the other.  Therefore, it is necessary to pass the arguments
      of the logarithmic terms in L_args.
  
      To handle the case where we are given Df/f, not f, use is_deriv_k_in_field().
  
      See also
      ========
      is_log_deriv_k_t_radical_in_field, is_log_deriv_k_t_radical
  
      """
      # Compute Df/f
      dfa, dfd = (
          fd * derivation(fa, DE) - fa * derivation(fd, DE)
      ), fd * fa
      dfa, dfd = dfa.cancel(dfd, include=True)
  
      # Our assumption here is that each monomial is recursively transcendental
      if len(DE.exts) != len(DE.D):
          if [i for i in DE.cases if i == "tan"] or (
              {i for i in DE.cases if i == "primitive"}
              - set(DE.indices("log"))
          ):
              raise NotImplementedError(
                  "Real version of the structure "
                  "theorems with hypertangent support is not yet implemented."
              )
  
          # TODO: What should really be done in this case?
          raise NotImplementedError(
              "Nonelementary extensions not supported "
              "in the structure theorems."
          )
  
      E_part = [
          DE.D[i].quo(Poly(DE.T[i], DE.T[i])).as_expr()
          for i in DE.indices("exp")
      ]
      L_part = [DE.D[i].as_expr() for i in DE.indices("log")]
  
      # The expression dfa/dfd might not be polynomial in any of its symbols so we
      # use a Dummy as the generator for PolyMatrix.
      dum = Dummy()
      lhs = Matrix([E_part + L_part], dum)
      rhs = Matrix([dfa.as_expr() / dfd.as_expr()], dum)
  
      A, u = constant_system(lhs, rhs, DE)
  
      u = u.to_Matrix()  # Poly to Expr
  
      if not A or not all(
          derivation(i, DE, basic=True).is_zero for i in u
      ):
          # If the elements of u are not all constant
          # Note: See comment in constant_system
  
          # Also note: derivation(basic=True) calls cancel()
          return None
      else:
          if not all(i.is_Rational for i in u):
              raise NotImplementedError(
                  "Cannot work with non-rational "
                  "coefficients in this case."
              )
          else:
              terms = [DE.extargs[i] for i in DE.indices("exp")] + [
                  DE.T[i] for i in DE.indices("log")
              ]
              ans = list(zip(terms, u))
              result = Add(*[Mul(i, j) for i, j in ans])
              argterms = [DE.T[i] for i in DE.indices("exp")] + [
                  DE.extargs[i] for i in DE.indices("log")
              ]
              l = []
              ld = []
              for i, j in zip(argterms, u):
                  # We need to get around things like sqrt(x**2) != x
                  # and also sqrt(x**2 + 2*x + 1) != x + 1
                  # Issue 10798: i need not be a polynomial
                  i, d = i.as_numer_denom()
                  icoeff, iterms = sqf_list(i)
                  l.append(
                      Mul(
                          *(
                              [Pow(icoeff, j)]
                              + [Pow(b, e * j) for b, e in iterms]
                          )
                      )
                  )
                  dcoeff, dterms = sqf_list(d)
                  ld.append(
                      Mul(
                          *(
                              [Pow(dcoeff, j)]
                              + [Pow(b, e * j) for b, e in dterms]
                          )
                      )
                  )
              const = cancel(
                  fa.as_expr() / fd.as_expr() / Mul(*l) * Mul(*ld)
              )
  
              return (ans, result, const)
  
imports: import itertools
  from functools import reduce
  from sympy.core.intfunc import ilcm
  from sympy.core import Dummy, Add, Mul, Pow, S
  from sympy.integrals.rde import order_at, order_at_oo, weak_normalizer, bound_degree
  from sympy.integrals.risch import gcdex_diophantine, frac_in, derivation, residue_reduce, splitfactor, residue_reduce_derivation, DecrementLevel, recognize_log_derivative
  from sympy.polys import Poly, lcm, cancel, sqf_list
  from sympy.polys.polymatrix import PolyMatrix
  from sympy.solvers import solve
label_desc: Use image node_id 17 for calling a global function with example usage: is_deriv_k(fa, fd, DE) and returns: None, ans,  result,  const