Solution
= Solution
The <total derivative> is the unique <linear map> $Df_a:\mathbb R^m\to\mathbb R^n$ satisfying
$$
\boxed{f(a+h)=f(a)+Df_a(h)+o(\|h\|)\quad\text{as }h\to0.}
$$
This describes the best first-order linear approximation without choosing coordinates. The <chain rule> is
$$
\boxed{D(g\circ f)_a=Dg_{f(a)}\circ Df_a.}
$$
The evaluation point $f(a)$ in the outer <derivative> matters, particularly when composing a quadratic <matrix> map with the exponential.