Some texts, such as Pavel Grinfeld's, talk about a $\delta/\delta t$-derivative whose role (in trajectory analysis of particles using tensor calculus) is pretty obscure to me. For example, the aforesaid text defines the acceleration $\mathbf{A}$ of a particle on a surface as
$$\mathbf{A} = \frac{\delta V^{\alpha}}{\delta t}\mathbf{S}_{\alpha} + \mathbf{N}B_{\alpha\beta}V^{\alpha}V^{\beta},$$
where
$$\frac{\delta V^{\alpha}}{\delta t} = \frac{d V^{\alpha}}{d t}+\Gamma^{\alpha}_{\beta\gamma}V^{\beta}V^{\gamma}.$$
I already know that the RHS of the second equation is the contravariant component of the acceleration vector $\mathbf{A}$. But, I don't get why $\frac{\delta V^{\alpha}}{\delta t}$, and not simply $\frac{d V^{\alpha}}{d t}$, is multiplied to $\mathbf{S}_{\alpha}$ in the first equation. In particular, given $\mathbf{V} = V^{\alpha}\mathbf{S}_{\alpha}$, one may write
$$\begin{align} \mathbf{A} &= \frac{d}{d t}[V^{\alpha}\mathbf{S}_{\alpha}]\\&= \mathbf{S}_{\alpha}\frac{d V^{\alpha}}{d t}+V^{\alpha}\frac{d \mathbf{S}_{\alpha}}{d t}\\&= \mathbf{S}_{\alpha}\frac{d V^{\alpha}}{d t} + V^{\alpha}\frac{d \mathbf{S}_{\alpha}}{d S^{\beta}}\frac{d S^{\beta}}{d t}\\&=\mathbf{S}_{\alpha}\frac{d V^{\alpha}}{d t} + \mathbf{N}B_{\alpha\beta}V^{\alpha}V^{\beta}. \end{align}$$
So,
1- can one explain what is wrong in my derivation above?
2- generally, why do we define $\delta/\delta t$-derivative?