Introduce formal derivatives in F[x]. Continuing the ideas of Exercise 15, shows that: a. D(af(x)) = aD(f(x))
Question:
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 +· · ·aixi · · ·anxn The derivatives of f'(x) is the polynomial f(x) = a0 + a1 +· · ·(i .1)aixi-1 +· · ·(n .1)anxn-1 , where i . 1 has its usual meaning for i ∈ Z+ and 1 ∈ F. These are formal derivatives; no "limits" are involved here.
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