Computing $DF(c),$ where $f:\Bbb R^n\to\Bbb R$ and $F:\Bbb R^n\times\Bbb R^n\to\Bbb R, F(x,y)=\cos(\langle f(x)\cdot y,f(y)\cdot x\rangle)$
Consider the scalar $a = f(\mathbf{x}) f(\mathbf{y}) \mathbf{x}^T \mathbf{y}$ It follows $$ dF = -\sin(a) da $$ It remains to compute \begin{eqnarray} da &=& [Df(\mathbf{x}) f(\mathbf{y}) \mathbf{x}^T \mathbf{y}+ f(\mathbf{x}) f(\mathbf{y}) \mathbf{y}] d\mathbf{x}+ [Df(\mathbf{y}) f(\mathbf{x}) \mathbf{x}^T \mathbf{y}+ f(\mathbf{x}) f(\mathbf{y}) \mathbf{x}] d\mathbf{y} \end{eqnarray}