Covariant derivative along a curve

See @lee2006riemannian Lemma 4.9.
How can we derive vector fields defined on a curve, with respect to the parameter of the curve?

Given a covariant derivative operator $\nabla$ in a manifold $M$ , for each curve $γ : I \to M$ , $\nabla$ determines a unique operator

D_{t} : T (γ) \to T (γ)

satisfying the following properties:
(a) Linearity over $R$ :

D_{t} (a V + b W) = a D_{t} V + b D_{t} W for a, b \in R .

(b) Product rule:

D_{t} (f V) = \dot{f} V + f D_{t} V for f \in C^{\infty} (I) .

D_{t} V (t) = \nabla_{\dot{γ} (t)} \tilde{V} .

For any $V \in T (γ)$ , $D_{t} V$ is called the covariant derivative of $V$ along $γ$ .

Local coordinate expression

D_{t} V (t_{0}) = {\dot{V}}^{i} (t_{0}) \partial_{i} + V^{i} (t_{0}) \nabla_{\dot{γ} (t_{0})} \partial_{i}

= ({\dot{V}}^{k} (t_{0}) + V^{i} (t_{0}) {\dot{γ}}^{j} (t_{0}) Γ_{i j}^{k} (γ (t_{0}))) \partial_{k} .