Derivative of a generic polynomial function Let $P:M_n(\mathbb{C})\to\mathbb{C}$ be a polynomial function, $A=(A_{ij}),E_{ji}\in M_n(\mathbb{C})$,where $E_{ji}$ is defined as matrix filled with zeros except $1$ in j-th row and i-th column  and $t\in \mathbb{R}$.
I want to take derivative of $P((I+tE_{ij})A)$ with respect to $t$ and evaluate it in $t=0$ afterwards.
My textbook suggests it is $$\frac{d}{dt}P((I+tE_{ji})A)|_{t=0}=\sum_{k,l}\frac{\partial P}{\partial A_{lk}}(A)\cdot (E_{ji}A)_{lk}.$$
My intuition says it makes use of chain rule, but I am failing to see where comes the sum from. 

Hint
You have 
$$A= \sum\limits_{k,l}A_{kl}E_{kl}$$
