(** * Partial derivatives *)

Definition partial_derive (m k : nat) (f : R -> R -> R) : R -> R -> R :=
  fun x y => Derive_n (fun t => Derive_n (fun z => f t z) k y) m x.Definition differential (p : nat) (f : R -> R -> R) (x y dx dy : R) : R :=
  sum_f_R0
    (fun m =>
      C p m *
      partial_derive m (p - m)%nat f x y *
      dx ^ m * dy ^ (p - m)%nat)
    p.Definition DL_pol (n : nat) (f : R -> R -> R) (x y dx dy : R) : R :=
  sum_f_R0
    (fun p =>
      differential p f x y dx dy / INR (fact p))
    n.

Lemma partial_derive_add_zero: forall f p q r s x y, (q=0)%nat \/ (r=0)%nat -> partial_derive p q (partial_derive r s f) x y = partial_derive (p+r) (q+s) f x y.