Guys, say we have a differentiable curve $\gamma\colon\mathbb R\to M_2\mathbb R$ and $\beta\colon M_2\mathbb R\to\mathbb R\colon A\mapsto (Ax,Ay)$ (for some fixed $x,y$), where $(\cdot,\cdot)$ is a bilinear map. I fail to show that
$$
\frac{d}{dt}_{t=0}\beta\gamma(t)=(\gamma'(t)x,\gamma(t)y)+(\gamma(t)x,\gamma'(t)y).
$$