Derivative of $2$nd order tensor is $3$rd order tensor (proof)

Question

I can see by intuition that, by taking the derivative of a tensor $A_{ij}$, you have $$A_{ij,\,k}=\frac{\partial A_{ij}}{\partial x_k}$$ which seems like a $3$rd order tensor with $27$ elements. How would a more formal proof look like?

Answer 1

Technically, you need to take something called the covariant derivative, not the partial derivative, if you want to get a tensor back. (However, depending on the metric tensors that raise and lower tensor indices, these two derivatives can be identical; it comes down to whether the Christoffel symbols vanish.) Whether you want to prove one works or the other doesn't, go back to the transformation law that defines a tensor. The point of the covariant derivative is to commute with metric tensors.