simplify fraction (stone drops off a cliff) How do I get from
$$ \frac {dt} {T} = \frac {dx} {gt} \sqrt { \frac {g} {2h} } $$
to
$$ \frac {dt} {T} = \frac {1} {2 \sqrt {hx} } dx $$
where $x(t) = \frac {1} {2} gt^2 $ and $ T = \sqrt { \frac {2h} {g}}$.
I'm currently struggling with Griffiths' Introduction to Quantum Mechanics. This is from a worked example on p.11-12 of the 2nd edition.

We have
$$
x=\frac{1}{2}gt^2
\quad\Leftrightarrow\quad
gt^2=2x
$$
so
\begin{align*}
\frac{dt}{T}
&
=\frac{dx}{gt}\sqrt{\frac{g}{2h}}
=\sqrt{\frac{g}{g^2t^2\cdot2h}}\,dx
=\sqrt{\frac{1}{gt^2\cdot2h}}\,dx
\\&
=\sqrt{\frac{1}{2x\cdot2h}}\,dx
=\sqrt{\frac{1}{4hx}}\,dx
=\frac{1}{\sqrt{4hx}}\,dx
=\frac{1}{2\sqrt{hx}}\,dx.
\end{align*}
