I have just started learning differential geometry and I am trying to answer the following exercise (it's not for homework):
Let $M$ be an $n$-dimensional manifold with chart $(U, \varphi_U)$ and let $(x^1, \dots, x^n)$ be the corresponding local coordinates. Define a vector field on $U$ by $X = X^i\frac{\partial}{\partial x^i}$. We will try to define a "derivative of vector fields" by $d_Y X = d_Y X^i \frac{\partial}{\partial x^i}$ By changing coordinates to $(\bar{x}^1, \dots, \bar{x}^n)$, show that unlike $df$, this operator is not independent of the choice of local coordinates.
I think I understand the method, but I am not certain of the details. I believe what I have to do is first apply $d_Y$ to $X$ and then change coordinates, and then change coordinates and apply $d_Y$ to $X$, and show that the two results are not then same. Here is what I have attempted:
Let $Y = Y^i\frac{\partial}{\partial x^i}$ (the question doesn't define $Y$ but presumably it's just another vector field). Then $d_Y X = d_Y X^i \frac{\partial}{\partial x^i}$
$= dX^i(Y) \frac{\partial}{\partial x^i}$
$= \frac{\partial X^i}{\partial x^j}Y^j \frac{\partial}{\partial x^i}$
This is just another vector field, so transforming it according to the contravariant transformation we get the following messy expression:
$\frac{\partial \bar{x}^i}{\partial x^j} \frac{\partial X^j}{\partial x^k} Y^k \frac{\partial}{\partial \bar{x}^i},$
which is $d_Y X$ in $\bar{x}^i$ coordinates (if I'm doing this correctly). Now I am trying to first change coordinates and apply $d_Y$ to $X$: first, in the new coordinates the vector field $X$ will be $\bar{X} = \frac{\partial \bar{x}^i}{\partial x^j}X^j \frac{\partial}{\partial \bar{x}^i}$ (contravariant transformation). So applying $d_Y$ to this we have $d_Y \bar{X} = d_Y \left(\frac{\partial \bar{x}^i}{\partial x^j}X^j \right) \frac{\partial}{\partial \bar{x}^i}$
Now I'm not sure what to do because it looks like $d_Y \left(\frac{\partial \bar{x}^i}{\partial x^j}X^j \right)$ will be terribly messy, and I don't really know how to evaluate it. Also, I don't know if I'm supposed to transform $Y$ to new coordinates as well (this would make the calculation even uglier) or if it's just $X$ that is transformed.
Could anyone help?