I'm trying to understand solution of the following problem from Landau, Lifshitz, Classical Theory Of Fields:

(ending skipped).

What I see when I "write out the expression for $w^iw_i$", using definition $w^i=\frac{du^i}{ds}=\frac{du^i}{cdt\sqrt{1-v^2/c^2}}$: $$\frac1{c^2-v^2}\left(\left(\frac{d}{dt}\frac1{\sqrt{1-v^2/c^2}}\right)^2-\left(\frac{d}{dt}\frac{v}{c\sqrt{1-v^2/c^2}}\right)^2\right)=-\frac{w^2}{c^4}$$

I then take the derivatives and arrive at the following result: $$-\frac{c^2\dot v^2}{(c^2-v^2)^3}=-\frac{w^2}{c^4}$$

I can get rid of the squares, use some guessing to determine correct signs, and then solve the resulting equation, getting correct results (according to the book). But the original solution in the book looks way simpler and must be better justified than mine.

So, the question: how did LL get their bottom-left equation? Did they "write out the expression for $w^iw_i$" in some other way than I did? How should I do here instead?