Let \((x^i)\) be local coordinates. We write the component functions of \(\gamma\) as \(\gamma(t)=(x^1(t),\ldots ,x^n(t))\). Then using the generalization of the chain rule for multi-variable functions, we obtain
\begin{equation*} (f\circ \gamma)'(t)=\partial_i f \, \dot{x}^i(t)= df_{\gamma(t)}(\gamma'(t))=\nabla_{\gamma'(t)} f. \end{equation*}