Understanding operator under a subtitution

With

$x = e^t \tag 1$

we have

$\dfrac{dx}{dt} = e^t = x; \tag 2$

thus for any function $f$

$x\dfrac{df}{dx} = \dfrac{dx}{dt} \dfrac{df}{dx} = \dfrac{df}{dt} \tag 3$

by the chain rule. Thus,

$x\dfrac{d}{dx} = \dfrac{d}{dt}. \tag 4$

$\frac {dy} {dt} =\frac {dy} {dx}\frac {dx} {dt} =\frac {dy} {dx} e^{t}=\frac {dy} {dx} x$.