Let $R$ be a ring and $R\left[ x \right]$ be the ring of polynomials over $R$.
If $f=a_nx^n+\cdots+a_0 \in R\left[ x \right]$, we define the formal derivative $f^{'}=na_nx^{n-1}+\cdots+a_1$.
Let $f$ , $g$ $\in R\left[ x \right]$ and $c \in R$. Using the definition, it can be verified that:
$(f+g)^{'}=f^{'}+g^{'}$
$(cf)^{'}=cf^{'}$
$(f+g)^{'}=f^{'}+g^{'}$
However, I came across this passage on Wikipedia:
There is a problem with this definition for noncommutative rings. The formula itself is correct, but there is no standard form of a polynomial. Therefore using this definition it is difficult to prove that $(f(x)\cdot b)^{'}=f^{'}(x)\cdot b$.
This makes me confused. I don't know what it is talking about. I don't see any problems here. The proofs of 1,2,3 for commutative rings and noncommutative rings are all the same. For example, I can easily prove that $(f\cdot b)^{'}=f^{'}\cdot b$:
Let $f_k$ denote the k-th coefficient of the polynomial $f$, $k=0,1,2\ldots$. By definition we know $(f^{'})_k=(k+1)f_{k+1}$. So $(LHS)_k=(k+1)(f\cdot b)_{k+1}=(k+1)(f_{k+1}\cdot b)=((k+1)f_{k+1})\cdot b=(f^{'})_k\cdot b=(f^{'}\cdot b)_k=(RHS)_k$. QED.
Where did I go wrong? What does the paragraph on Wikipedia mean?
Any insights are much appreciated.