I'm confused on how I am supposed to differentiate the following using the chain rule.
I have $x(t)$ and I have defined new time variables $t=\tau$ and $T=\epsilon t$
I have that
$$ \frac{dx}{dt} = \frac{d \tau}{dt} \frac {\partial x}{\partial \tau} + \frac {dT}{dt} \frac {\partial x}{\partial T} $$
which becomes
$$ \frac {dx}{dy} = \frac {\partial x}{\partial \tau} + \epsilon \frac {\partial x}{\partial T} $$
Then I want to differentiate with respect to $t$, so
$$ \frac {d^2x}{dt^2}= \frac {d}{dt} \left(\frac {\partial x}{\partial \tau}\right)+\epsilon \frac {d}{dt}\left(\frac {\partial x}{\partial T}\right)$$
but I am unsure on how to compute the last part with workings. I have the answer to it but I am unsure on what the steps are and how to differentiate the parital derivatives.
Thanks in advance