Possibly very simple chain rule application.
I have an unknown function $u(x)$, I make a change of variables $x=t^2/4$ where $dx/dt=t/2$
I want to find now $u''(t)$ and $u'(t)$ (in terms of $u''(x)$ and $u'(x)$
So far I have found the first derivative
$$\frac{du}{dt}=\frac{du}{dx}\frac{dx}{dt}=\frac{du}{dx}\frac t 2=u'(t)$$
However I am unsure how to apply chain rule to expand this to a second derivative. Many thanks
We have that in short notation by chain rule and product rule
$u_t=u_x\cdot x_t$ $$\implies \frac{du}{dt}=\frac{du}{dx}\frac{dx}{dt}$$
$u_{tt}=(u_x\cdot x_t)_t=u_{xx}\cdot x_t^2+u_x\cdot x_{tt}$$$\implies \frac{d^2u}{dt^2}=\frac{d^2u}{dx^2}\left(\frac{dx}{dt}\right)^2+\frac{du}{dx}\frac{d^2x}{dt^2}$$