Formal derivative of a polynomial

Definition

Suppose $K$ is a field and $f(x)\in K[x]$ is a polynomial. The formal derivative of $f$, denoted $f^{\prime }$, is defined as follows.

If $f(x)=\sum _{m=0}^n{a}_m{x}^m$, then $f^{\prime }(x)=\sum _{m=1}^{n}m{a}_m{x}^{m-1}$.

Here, the $m{a}_m$ is understood as $a_m$ added to itself $m$ times.

Facts

The formal derivative gives a $K$-linear map from $K[x]$ to itself. When $K$ has characteristic zero, the kernel of the map is $K$, whereas when $K$ has characteristic $p$, the kernel of the map is $K[x^p]$.

The formal derivative also satisfies the following rule for multiplication, called the Leibniz rule:

$(fg{)}^{\prime }(x)=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$.