4) Let F be a field. The derivation f' (T) € F[T] of a polynomial f(T) E a; · Ti e F[T] is defined by i20 f'(T) := E(i- a;) - T*-1. i21 Verify for all f(T) and g(T) in F[T] the rules (F(T) + g(T))' = f'(T) +g'(T) and (f(T) · g(T))' = f'(T) · g(T) + f(T) · gʻ(T).
4) Let F be a field. The derivation f' (T) € F[T] of a polynomial f(T) E a; · Ti e F[T] is defined by i20 f'(T) := E(i- a;) - T*-1. i21 Verify for all f(T) and g(T) in F[T] the rules (F(T) + g(T))' = f'(T) +g'(T) and (f(T) · g(T))' = f'(T) · g(T) + f(T) · gʻ(T).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![4) Let F be a field. The derivation f'(T) e F[T] of a polynomial f(T) =
Ea; · Ti e F[T] is defined by
i20
f'(T) := E(i · a;) - Ti-1.
i21
Verify for all f(T) and g(T) in F[T] the rules
(F(T)+ g(T))' = f'(T) + g'(T)
and
(F(T) · g(T))' = f'(T) · g(T) + f(T) · gʻ(T).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1bbb03a-1330-4abd-86a4-84230eb34f64%2F05af52f9-a3c9-4805-b607-8b5f17a7f858%2Fyxi2vml_processed.png&w=3840&q=75)
Transcribed Image Text:4) Let F be a field. The derivation f'(T) e F[T] of a polynomial f(T) =
Ea; · Ti e F[T] is defined by
i20
f'(T) := E(i · a;) - Ti-1.
i21
Verify for all f(T) and g(T) in F[T] the rules
(F(T)+ g(T))' = f'(T) + g'(T)
and
(F(T) · g(T))' = f'(T) · g(T) + f(T) · gʻ(T).
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