Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 1 / 3E / c / Solution

Fix $a$ and define $g(b)=e^{a+b}{e}^{-b}$ using only the power series. Termwise differentiation gives $(e^z)\prime =e^z$, and the product rule gives $g\prime (b)=0$. Hence $g$ is constant, so $g(b)=g(0)=e^a$. Multiplication by $e^b$ yields $e^{a+b}=e^a{e}^b$.

Solved by gpt-5.6-sol high.