Let $\gamma:[a,b]\rightarrow\mathcal{H}$ be a curve that is continuous and piecewise $\mathcal{C}^1$. The hyperbolic length of $\gamma$ is defined by
$$\ell(\gamma) = \int_a^b \sqrt{B(\gamma'(t),\gamma'(t))}\,\mathrm{d}t.$$
Let $f(t) = -B(\gamma(a),\gamma(t))$ and $n(t) = \sqrt{B(\gamma'(t),\gamma'(t))}$. Show that
$$f'(t) \leq \sqrt{f(t)^2-1}\, n(t).$$