= Solution
{c}
The <Fréchet derivative> at $a$ is the <linear map> $Df_a:\mathbb R^n\to\mathbb R^k$ satisfying
$$
f(a+h)=f(a)+Df_a(h)+r(h),\qquad
\frac{\|r(h)\|}{\|h\|}\longrightarrow0.
$$
It is unique. When both derivatives exist, the <chain rule> is $D(g\circ f)_a=Dg_{f(a)}\circ Df_a$.
For the square map,
$$
(A+\Delta)^2-A^2=A\Delta+\Delta A+\Delta^2.
$$
The remainder has norm at most $\|\Delta\|^2$ in a submultiplicative <matrix norm>, hence is $o(\|\Delta\|)$. Therefore
$$
\boxed{DS_A(\Delta)=A\Delta+\Delta A.}
$$
<Matrix> multiplication is not commutative, so the two terms cannot generally be combined as $2A\Delta$.
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