The definition of $K_1$ is stated in "The K-book" by Charles Weibel as a quotient of $GL(R)$, where $GL(R)$ is the union of the sequence $R^{ \times} = GL_1(R) \subset GL_2(R) \subset ...\subset GL_n(R) \subset GL_{n+1}(R) \subset..$ and $K_1(R)$ is defined as $GL(R)/[GL(R),GL(R)].$ Right now I am stuck in the computation of $K_1(k[x]/(x^2))$, for a field $k$. The sudden interest to this is due to the example mentioned in the book that for every field $k$, $K_1(k) = k^{\times}$. So out of curiosity I experimented and understood that $K_1((k[x]/(x^2))_{red}) \cong k^{\times}.$ But got stuck in the computation of my headline question. Any hints or way to the solution are really appreciated. Thanks in advance.

$\DeclareMathOperator\GL{GL}$The definition of $K_1$ is stated in "The K-book" by Charles Weibel as a quotient of $\GL(R)$, where $\GL(R)$ is the union of the sequence $R^{ \times} = \GL_1(R) \subset \GL_2(R) \subset \dotsb\subset \GL_n(R) \subset \GL_{n+1}(R) \subset\dotsb$ and $K_1(R)$ is defined as $\GL(R)/[\GL(R),\GL(R)]$. Right now I am stuck in the computation of $K_1(k[x]/(x^2))$, for a field $k$. The sudden interest to this is due to the example mentioned in the book that for every field $k$, $K_1(k) = k^{\times}$. So out of curiosity I experimented and understood that $K_1((k[x]/(x^2))_\text{red}) \cong k^{\times}$. But I got stuck in the computation of my headline question. Any hints or way to the solution are really appreciated.