Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 126 / 1 / ii / Solution

Every $k$-derivation $D:K[t]\to M$ is uniquely determined by its restriction to $K$ and by $D(t)$. Conversely, a $k$-derivation $K\to M$ and an arbitrary element $m\in M$ extend uniquely byRepresenting this natural decomposition of derivations gives