I am trying to prove that $T_e(U(n))\subset \mathfrak{u}(n)$ and for this I am taking $X\in T_e(U(n))$ and I need to prove that $X+X^*=0$. As $X\in T_e(U(n))$ then $X=\dot\gamma(0)$ for some $\gamma:I\to U(n)$, then $$X+X^*=\frac{d}{dt}|_{t=0}\gamma(t)+(\frac{d}{dt}|_{t=0}\gamma(t))^*=\frac{d}{dt}|_{t=0}(\gamma(t)+\gamma(t)^*)=\frac{d}{dt}|_{t=0} 0=0$$
but I am not sure if $(\frac{d}{dt}|_{t=0}\gamma(t))^*=\frac{d}{dt}|_{t=0}\gamma(t)^*$, could someone help me to say if what I do is correct or if there is any other way that is correct? Thank you!