formula

$L(t_0+dt)-L(t_0)&=&\sqrt{\frac{t_0+dt}{L_0}}-\sqrt{\frac{t_0}{L_0}}\\ &=& \sqrt{\frac{t_0}{L_0}}\left(\sqrt{1+\frac{dt}{t_0}}-1\right)\\ &\simeq& \sqrt{\frac{t_0}{L_0}}\cdot\frac{1}{2}\frac{dt}{t_0}\\ &=& \frac{dt}{2\sqrt{t_0L_0}}$