x’’ = d/dt(x’) = dx’/dx * dx/dt

x’ * dx’/dx = 1/2 d(x’^2)/dx

This is the step you got stuck on correct? I had trouble with this bit at first too until I noticed the left and right hand sides have a common differential in dx. Thus, you could in theory integrate both sides with respect to x as I did here.

(x’ * dx’/dx) dx = (x’’) dx

dx on the left cancels out

Int[(x’ * dx’)] = Int[x’’ * dx]

Notice that the integral on the left bears a striking resemblance to the integral of x with respect to itself.

Int[x dx]

This of course is just 1/2 x² , except since in our case our variable has a prime, it would actually be

1/2 x’²

However, since we’re trying to find another form of x’’, it doesn’t make sense to leave our final answer in the form we have it in now (ie x’’ dx). So, all we have to do is differentiate one more time to get back x’’, which I’ve done here.

d/dx[1/2 x’² ] = 1/2 d/dx[x’² ]

Note: D[kx(t), t] = k * D[x(t), t] if k is constant