Two random scalar variables, x and y, are related by the following expression: $y(t) = f[x(t)] = a\frac{dx(t)}{dt}$, where $x(t)$ is periodic and $a$ is a constant.
How can I calculate the pdf $g_{Y}[y(t), t]$, knowing the pdf $g_{X}[x(t), t]$?
I'm confusing myself on how to apply $g_{Y}(y, t)=g_{X}(x=f^{-1}(y), t)\left|\frac{dx}{dy}\right|_{y=y}$