Start with this:
$\frac{\partial f}{\partial x'^i} = \frac{\partial f}{\partial x^j} \frac{\partial x^j}{\partial x'^i}$
I think(?) the $\partial x^j$s cancel and this simplifies to $\frac{\partial f}{\partial x'^i}$.
However if one writes out the summation explicitly, substituting 1,2,3 for j (in 3 dimensions),
$\frac{\partial f}{\partial x^j} \frac{\partial x^j}{\partial x'^i} = \frac{\partial f}{\partial x^1} \frac{\partial x^1}{\partial x'^i} + \frac{\partial f}{\partial x^2} \frac{\partial x^2}{\partial x'^i} + \frac{\partial f}{\partial x^3} \frac{\partial x^3}{\partial x'^i} = 3 \frac{\partial f}{\partial x'^i}$
I think the 3 is not right. Why not?
I am the master of stupid questions.