How can we prove that the exponential function shifts an arbitrary non-chaotic differential operator by a constant?

Question

if $f(D)y$ is an arbitrary differential operator, of the basic differential $D \equiv \frac{d}{dx}$, acting on any generic infinitely differentiable function $y(x)$ of some real independent variable $x \in \mathbb{R} $, how can we prove (or disprove) that

$$f(D) e^{ax}y(x) = e^{ax}f(D+a)y(x)$$ $$\therefore f(D+a)y(x) =e^{-ax} f(D) e^{ax}y(x)$$?