in my notes, I have the following phrase:
With $x = e^t$ and $x \frac{d}{dx} = \frac{d}{dt}$
How, how come we get $x \frac{d}{dx} = \frac{d}{dt} $? I know if I diferentiate with respect to $t$ I obtain
$$ \frac{d}{dt} x = x $$
I do not understand how this operators are defined? maybe I am misunderstading the notation?
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,
$x\dfrac{d}{dx} = \dfrac{d}{dt}. \tag 4$
$\frac {dy} {dt} =\frac {dy} {dx}\frac {dx} {dt} =\frac {dy} {dx} e^{t}=\frac {dy} {dx} x$.
