Problem
ALG-B3-M10-P004 Rational Derivation Form
#4
★★★★☆ Level 4 of 5
Let \(f:\mathbb Q\to\mathbb Q\) be additive and satisfy \(f(xy)=xf(y)+yf(x)\). Find \(f\).
Reduce to \(f(x)=cx\).
Additivity on \(\mathbb Q\) gives \(f(x)=cx\). Then \(cxy=2cxy\) for all \(x,y\). Hence \(c=0\). The answer is \(f\equiv0\).
The system kills the linear coefficient.