derive()

For a set A ⊆ Rⁿ(or a banach space(?)) we define

the differential operator

D=∂/∂x:(A→R^m)→(A→Hom(Rⁿ,R^m)) ⊂ (A→(Rⁿ→R^m)); f ↦ Df
where

Df:A→Hom(Rⁿ,R^m); x ↦ (Df)(x),
(Df)(x):Rⁿ→R^m; h ↦ (Df)(x)(h) = f'(x)·h with f'(x)∈R^m×n (which is the functional matrix)

If f is derivable at x∈Rⁿ it is (f(x+h) - f(x) - (Df)(x)(h)) / ||h|| → 0 (h→0)

the total derivative

df/dx = f' = ∂f/∂x = (∂fi/∂xj)i,j =	(	∇f1	)	=	[	∂f1/∂x1,	∂f1/∂x2, ,	…,	∂f1/∂xn	]
		∇f2				∂f2/∂x1,	∂f2/∂x2,	…,	∂f2/∂xn
		⋮				⋮		…	⋮
		∇fm				∂fm/∂x1,	∂fm/∂x2,	…,	∂fm/∂xn

The total derivative df/dx:A→R^m×n can be identified with the matrix of Df. Note that we identify a matrix of real-valued unary functions with a matrix-valued function, here. If it exists it is identical to the functional matrix or Jacobi matrix of the partial derivatives ∂f_i/∂x_j. Note that if we identify Hom(Rⁿ,R^m) = R^m×n it is true that f' = Df.

the total differential

(Df)(x)(dx) = f'(x)·dx = ∂f/∂x(x)·dx = ∂f/∂x₁(x)dx₁ + ∂f/∂x₂(x)dx₂ + … + ∂f/∂x_n(x)dx_n for h:=dx∈Rⁿ.