Derivative of matrices to power 2

Question

For $f(A) = A^2$, $Mat(2,2) \rightarrow Mat(2,2)$, why its derivative is $[DS(A)]B=AB+BA$? I found an answer here Is there a general form for the derivative of a matrix to a power?, but after read it I still don't understand. Thanks!

Answer 1

Compute $[DS(A)]B$ as the directional derivative $$\lim_{t\to 0} \frac{S(A+tB)-S(A)}t = \lim_{t\to 0} \frac{(A^2+tAB+tBA+t^2B^2) - A^2}t.$$