I know that the $(i,k)$ element of a matrix product is expressed in tensor notation as:
$$M^i_{\:k} = A^i_{\:j}B^j_{\:k}$$
But I'm a little confused as to what happens in the notation when the product involves more than two matrices. $$Q^i_{\:k} = A^i_{\:j}B^?_{\:?} C^?_{\:?}$$
Also, say the elements of $A$ and $B$ are functions of corresponding elements of a vector $x$.
$$A^i_{\:j} = f(x_i, x_j)$$
$$B^i_{\:j} = g(x_i, x_j)$$
Same function $f()$ for all entries in $A$, and likewise same function $g()$ for all entries in $B$, just with different variables $x_i, x_j$ corresponding to the matrix indices.
How to represent differentiation of $M^i_{\:k}$ and $Q^i_{\:k}$ w.r.t. $x_i$
$$\frac{\partial M^i_{\:k}}{\partial x_i}=?$$ $$\frac{\partial Q^i_{\:k}}{\partial x_i}=?$$
The trick is $$Q^i{}_k=A^i{}_jB^j{}_rC^r{}_k.$$
For the differentiation one uses the Leibniz's Rule: $$\frac{\partial M^i{}_k}{\partial x^u}= \frac{\partial A^i{}_j}{\partial x^u}B^j{}_k +A^i{}_j\frac{\partial B^j{}_k}{\partial x^u},$$ and then take $u=i$ for the contraction.