Introduce formal derivatives in F[x].

Continuing the ideas of Exercise 15, shows that:

a. D(af(x)) = aD(f(x)) for all f(x) ∈ F[x] and a ∈ F.

b. D(f(x)g(x)) = f(x)g'(x) + ƒ'(x)g(x) for all f(x), g(x) ∈ F[x]. 

c. D((ƒ(x))^m) = (m · 1) ƒ (x)^m−¹ ƒ'(x) for all f(x) ∈ F[x]. 

Data from Exercise 15

Let F be any field and let f(x) = a₀ + a₁x +· · ·a_ixⁱ · · ·a_nxⁿ The derivatives of f'(x) is the polynomial f(x) = a₀ + a₁ +· · ·(i .1)a_ix^i-1 +· · ·(n .1)a_nx^n-1 , where i . 1 has its usual meaning for i ∈ Z⁺ and 1 ∈ F. These are formal derivatives; no "limits" are involved here.