How to show that the function $f:M_n(\mathbb R) \times M_n(\mathbb R) \to M_n(\mathbb R)$ given by $f(A,B)=AB$ is differentiable and how to find the total differential at a point $(X,Y)$ i.e. how to find $D f_{(A,B)}(X,Y)$ ?
2026-04-08 18:41:10.1775673670
To show that the map $f:M_n(\mathbb R) \times M_n(\mathbb R) \to M_n(\mathbb R)$ given by $f(A,B)=AB$ is differentiable and evaluate the derivative
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Hint: let be $h,k\in M_n(\Bbb R)$ "small" $$(A+h)(B+k) = AB + Ak + Bh + hk$$ And the linear term is...