Exercise 7.2.8 (Formal derivative)

Question

Check one or two of the properties in Lemma 7.2.7.

Hassaan Naeem · Accepted Answer

extbf{Solution:}
We check the additive property.\\
We let $f(t) = \sum_{i=0}^{n} a_it^i \in K[t]$ and $ g(t) = \sum_{i=0}^{n} b_it^i \in K[t]$.
Then we have that $f(t)+g(t) = \sum_{i=0}^{n} (a_i+b_i)t^i = \sum_{i=0}^{n} c_it^i \in K[t]$, where
$c_i = (a_i+b_i)$. Then by 	extbf{Definition 7.2.6} we have that $D(f+g)(t) = \sum_{i=1}^{n} ic_it^{i-1} = \sum_{i=1}^{n}i(a_i+b_i)t^{i-1} = \sum_{i=1}^{n} ia_it^{i-1} + \sum_{i=1}^{n} ib_it^{i-1} = Df + Dg \in K[t]$

Exercise 7.2.8 (Formal derivative)

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Exercise 7.2.8 (Formal derivative)

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