I need to show that $\frac{\partial}{\partial x^i}v^{j}(x)$ is not a tensor.
Here $v^i(x)$ is a vector field. I tried writing down the tensor transformation like this:
$$\frac{\partial}{\partial{x'^{i'}}}v'^{j'}(x')=\frac{\partial x^i}{\partial x'^{i'}} \frac{\partial x'^{j'}}{\partial x^j} \frac{\partial}{\partial x^i}v^{j}(x)$$
And thus I think this looks like a tensor transformation is supposed to look. I've likely done something incorrectly, could someone please give me a hint here on what I'm doing wrong?
What you went wrong is that you don't write the derivation correctly. What you must do is this
$$ \frac{\partial v^{j'}}{\partial x^{i'}} = \frac{\partial}{\partial x^{i'}} v^{j'} = \underbrace{\frac{\partial x^i}{\partial x^{i'}} \frac{\partial}{\partial x^i}}_\text{by chain rule} \underbrace{\Bigg( \frac{\partial x^{j'}}{\partial x^j} v^j \Bigg) }_\text{by transf.rule for $v^{j'}$} = \dots (\text{proceed}) $$ which is by chain rule and tensor transformation rule $\textbf{for}$ $v^{j'}$. Proceed by yourself and find out why its not a tensor.