Let $X$ be a $\mathbb{R}$-valued heavy-tailed and continuous random variable with density function $p_X$. Furthermore, let $f:\mathbb{R}\to\mathbb{R}$ be continuously differentiable.
I am trying to show whether the following assertion is true:
If $X$ and $f(X)$ are heavy-tailed, so is \begin{equation} \left|f'(X) \ \cdot \ f(X)\right|. \end{equation}
To prove heavy-tailedness we need to show that for all $t>0$
\begin{equation} \mathbb{E}\left[\exp\left(t \cdot \left|f'(x) \ \cdot \ f(x)\right|\right) \right] = \int_{\mathbb R} \exp\left(t \cdot \left|f'(x) \ \cdot \ f(x)\right|\right) \ p_X(x) \ dx = \infty \end{equation}
Any hints?