Why is this step in proving that a partial derivative of a vector is not a tensor illegal? If I want to prove that the derivative of a vector $V$, $\partial_\mu V^\nu$, is not a tensor, I can apply the tensor transformation rule to $V^\nu$:
$$\partial_\mu V^\nu \rightarrow \partial_{\mu\prime} V'^{\nu\prime} = \partial_{\mu\prime}\left( \frac{\partial x'^{\nu'}}{\partial x^\nu}V^\nu\right)$$
Now the next correct step would be to apply the chain rule to $\partial_{\mu'}=\frac{\partial x^\mu}{\partial x'^{\mu'}}\frac{\partial}{\partial x^\mu}$ and then expand. Everything from this point is clear to me.
But my course notes mention that it would not be correct to differentiate the expression in the brackets with $\partial_{\mu\prime}$ right away, but I do not understand why.
Why would that step be illegal? Why did we have to apply the chain rule first?
 A: The shorthand notation is tripping you up.  Let's say that in a particular coordinate chart $x$, your vector components take the form $V^\mu (x)$.  You now decide to differentiate these components, to obtain an expression
$$\frac{\partial}{\partial x^\nu}\big[ V^\mu(x)\big]$$
Now you wish to change to a different coordinate chart $y$.  The components of $V$ transform as
$$V^\mu(x) \mapsto J^\mu_{\ \ \alpha}(y)  \hat V^\alpha(y)$$
where $J^\mu_{\ \ \alpha}(y)\equiv \frac{\partial x^\mu (y)}{\partial y^\alpha}$ is the Jacobian of the transformation.  Plugging that in yields
$$\frac{\partial}{\partial x^\nu}\left[J^\mu_{\alpha}(y)\hat V^\alpha(y)\right]$$
The problem now is that the quantity inside the brackets is a function of the coordinates $y$, so you cannot differentiate them with respect to the coordinates $x$.  Instead, you need to transform your derivative operator:
$$\frac{\partial}{\partial x^\nu} \mapsto \big(J^{-1}(y)\big)^\beta_{\ \ \nu} \frac{\partial}{\partial y^\beta}$$
Now you can expand everything out using the product rule. To recover the shorthand notation, you can drop the arguments $(x)$ and $(y)$ from all of the functions, write the Jacobian as $\partial x/\partial y$, and replace $y$ with $x'$. From there, replace $\frac{\partial}{\partial x^\mu}$ with $\partial_\mu$ and $\frac{\partial}{\partial x'^\mu}$ with $\partial_{\mu'}$.  (Can you see how this gets confusing for beginners?)

As an example, we can transform the vector field $\mathbf V = \frac{x}{\sqrt{x^2+y^2}}\frac{\partial}{\partial x}+\frac{y}{\sqrt{x^2+y^2}}\frac{\partial}{\partial y}$ from Cartesian coordinates $(x,y)$ to polar coordinates $(r,\theta)$. Clearly this is simply equal to $\frac{\partial}{\partial r}$, so this is what we should expect to find at the end.
Because $x(r,\theta)=r\cos(\theta)$ and $y(r,\theta) = r\sin(\theta)$, the Jacobian and its inverse take the form
$$J(r,\theta) = \pmatrix{\cos(\theta) & -r\sin(\theta)\\ \sin(\theta) & r\cos(\theta)} \qquad J^{-1}(r,\theta) = \pmatrix{\cos(\theta) & \sin(\theta)\\ -\frac{1}{r}\sin(\theta) & \frac{1}{r}\cos(\theta)}$$
The initial Cartesian components are $V^x(x,y) = \frac{x}{\sqrt{x^2+y^2}}$ and $V^y(x,y) = \frac{y}{\sqrt{x^2+y^2}}$. Expressed in terms of $r$ and $\theta$, we have that $V^x = \cos(\theta)$ and $V^y = \sin(\theta)$.  According to the transformation rule,
$$\pmatrix{V^x\\V^y} = J(r,\theta) \pmatrix{\hat V^r\\ \hat V^\theta} \implies \pmatrix{\hat V^r\\ \hat V^\theta} = J^{-1}(r,\theta) \pmatrix{\cos(\theta)\\\sin(\theta)} = \pmatrix{1\\0}$$
And so $\hat V^r(r,\theta) = 1$ and $\hat V^\theta(r,\theta) = 0$, as anticipated.
In this language, the question in the OP relates to expressions of the form
$$\frac{\partial}{\partial x} \pmatrix{V^x(x,y)\\ V^y(x,y)} = \frac{\partial}{\partial x} \left[ J(r,\theta) \pmatrix{\hat V^r(r,\theta)\\\hat V^\theta(r,\theta)}\right]$$
and the fact that it is inappropriate to differentiate the RHS before transforming the derivative operator.
A: What you write $A_{\mu}^{\nu} = \partial_{\mu} V^{\nu}$ is the component representation of the tensor
$\mathbb{A} = \mathbf{b}^{\mu}\otimes\mathbf{b}_{\nu} A_{\mu}^{\nu} = \mathbf{b}^{\mu}\otimes\mathbf{b}_{\nu} \partial_{\mu} V^{\nu}$,
assuming $\mathbf{b}_{\nu}$ as the base vectors you're referring to, and $\mathbf{b}^{\mu}$ the vectors of the reciprocal basis, s.t. $\mathbf{b}_{\nu} \cdot \mathbf{b}^{\mu} = \delta^{\mu}_{\nu}$.
We can write the components of your tensor in a different basis, whose vectors $\mathbf{b}'_{\alpha}$ can be as a linear combination of the vectors of the original basis, using the
$\mathbf{b}'_{\alpha} = T_{a}^{\nu} \mathbf{b}_{\nu}$$\qquad , \qquad$$\mathbf{b}'^{\beta} = \tilde{T}^{\beta}_{\mu} \mathbf{b}^{\mu}$
$\mathbf{b}_{\nu} = \tilde{T}^{a}_{\nu} \mathbf{b}'_{\alpha}$$\qquad , \qquad$$\mathbf{b}^{\mu} = {T}_{\beta}^{\mu} \mathbf{b}'^{\beta}$  ,
being $\tilde{T}^{\beta}_{\mu}$ the inverse matrix $ T_{a}^{\nu}$. Replacing the expression of the vectors of the basis in the expression of the tensor $\mathbb{A}$ we get
$\mathbb{A} = \mathbf{b}^{\mu}\otimes\mathbf{b}_{\nu} A_{\mu}^{\nu} 
 = \mathbf{b}'^{\beta}\otimes\mathbf{b}'_{\alpha} \underbrace{ {T}_{\beta}^{\mu} \tilde{T}^{a}_{\nu} A_{\mu}^{\nu}}_{{A'_{\beta}}^{\alpha}} = \mathbf{b}'^{\beta}\otimes\mathbf{b}'_{\alpha} {A'_{\beta}}^{\alpha}$
