# Flat function/Polynomial in 1 over x times exp -1 over x/Derivative/Exercise

Let ${\displaystyle {}p\in \mathbb {R} [Y]}$ be a polynomial and
${\displaystyle g\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto g(x)=p({\frac {1}{x}})e^{-{\frac {1}{x}}}.}$
Prove that the derivative ${\displaystyle {}g'(x)}$ has also the shape
${\displaystyle {}g'(x)=q({\frac {1}{x}})e^{-{\frac {1}{x}}}\,,}$
where ${\displaystyle {}q}$ is a polynomial.