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

Let ${}p\in \mathbb {R} [Y]$ be a polynomial and

$g\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto g(x)=p{\left({\frac {1}{x}}\right)}e^{-{\frac {1}{x}}}.$

Prove that the derivative ${}g'(x)$ has also the shape

${}g'(x)=q{\left({\frac {1}{x}}\right)}e^{-{\frac {1}{x}}}\,,$

where ${}q$ is a polynomial.

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