Covariant derivative in $\mathbb{R}^n$ I am studying my lecture notes on covariant derivative, and is having difficulty to do a computation: 

Suppose $X,Y$ be smooth vector fields in $\mathbb{R}^n.$ Consider the integral curve $c_p:(-\epsilon,\epsilon) \to \mathbb{R}^n$ of $X$ passing through $p\in \mathbb{R}^n$ at time $t=0$. We have an identification $T_{c_p(t)}\mathbb{R}^n \to T_p\mathbb{R}^n$ of the tangent spaces. Using this, we define the first order variation of the map $t \mapsto Y(c_p(t))$ to be
$$
\nabla_X(Y)(p):= \lim_{t\to 0} \frac{Y(c_p(t))-Y(p)}{t}.
$$

It`s OK so far. But I have problem understanding what my note says further: 

Writing $X=(X_1,\cdots,X_n), \ Y=(Y_1,\cdots,Y_n),$ one verifies that 
  $$
\nabla_X(Y)(p)=\bigg( \cdots, \sum_ {i=1}^n \frac{\partial Y_j (p)}{\partial x_i}X_i(p),\cdots\bigg).
$$

I don`t understand why the last expression is true. 
I have tried to do the computation and is stuck here: 
$$
\nabla_X(Y)(p)= \lim_{t\to 0}\frac{\sum_{i=i}^n (Y_i(c_p(t))-Y_i(p)).\frac{\partial}{\partial x_i}}{t}.
$$
I don`t know how to proceed from above step. Please help. 
 A: Consider, $\frac{d}{dt}Y(c(t)) = (\frac{d}{dt}Y_1(c(t)),  \cdots , \frac{d}{dt}Y_n(c(t)))$. Moreover, each component function of $Y$ is a function of the coordinates $x_1, \dots , x_n$:
$$ Y_j = Y_j(x_1, \dots , x_n )$$
Also, $c'(0) = X(p)$ by construction ($c$ is the curve whose tangent vector produces the vector attached to $p$ by the vector field $X$). Notice, this means that the $i$-th component of $c'(0)$ is given by $(x_i \circ c)'(0) = X_i$. Finally, the following is simply the multivariate chain rule:
$$ \frac{d}{dt} Y_j(c(t))\bigg{|}_0 = \sum_{i=1}^n \frac{\partial Y_j}{\partial x_i}(c(0)) \frac{d(x_i \circ c)}{dt}(0) = \sum_{i=1}^n \frac{\partial Y_j}{\partial x_i}(p) X_i $$
Notice, $X = (X_1, \dots, X_n)$ suggests the use of notation $(1,0,\dots, 0) = \frac{\partial}{\partial x_1}\big{|}_p$. If I was to continue with your direct calculation, my next move would be to use the result of advanced calculus which says that a vector of limits is the limit of the vector. In other words, you can take the limit component-wise. After that, you have to face the chain-rule for functions of $n$-variables.
