What am I missing here? That's an idiot question, but I'm missing something here. If $x'= Ax$ and $A$ is linear operator in $\mathbb{R}^n$, then $x'_i = \sum_j a_{ij} x_j$ such that $[A]_{ij} =a_{ij} = \frac{\partial x'_i}{\partial x_j}$, therefore $\frac{\partial}{\partial x_i'} = \sum_j \frac{\partial x_j}{\partial x'_i} \frac{\partial}{\partial x_j}$. However $\frac{\partial}{\partial x_i'} = \frac{\partial}{\partial \sum_j a_{ij} x_j} = \sum_j a_{ij} \frac{\partial}{\partial x_j} =  \sum_j \frac{\partial x'_i}{\partial x_j} \frac{\partial}{\partial x_j}$ !!! What's wrong here?
Thanks in advance.

When you wrote $\frac{\partial}{\partial x_i'} = \frac{\partial}{\partial \sum_j a_{ij} x_j} = \sum_j a_{ij} \frac{\partial}{\partial x_j} $ the $a_{ij}$ somehow climbed from the denominator to the numerator
