Derivative of matrix valued function $f(A)=AA^T$

Question

This is part of a larger proof on the orthonormal group viewed as a manifold. I have never done anything with matrix calculus and am trying to find $df$ where $$ f(A)=AA^T $$ where $$ f:M_{n\times n}\to S(n) $$ wehre $S(n)$ are the symmetric $n\times n$ matrices. So we should have $$ df:M_{n\times n}\to T_pS(n) $$ where $p\in S(n)$.

Answer 1

Hint: If you already know that $df$ exists, you can compute it as the directional derivative $$ df_A(V) = \lim_{t \to 0} \frac{f(A+tV) - f(A)}{t} = \lim_{t \to 0} \frac{(A+tV)(A+tV)^T - A A^T}{t} = \cdots $$