I want to show that $U(n)$ is a regular submanifold of $GL(n,\mathbb C)$, for this I figured I use the regular level set theorem. For $U(n)$ is the preimage of the identity $I$ under the map $f:GL(n,\mathbb C)\to GL(n,\mathbb C)$ defined by $A\mapsto A^*A$. So I want to show that the differential of $f$ at $A\in U(n)$ is surjective. For this let $X\in T_AGL(n,\mathbb C)$, so that we identify $X$ with a matrix in $\mathbb C^{n\times n}$. Let $\gamma(t)=A\exp(tA^*X)$, so that $\gamma(0)=A$ and $\gamma'(0)=AA^*X\exp(0\cdot A^*X)=X$. Now we compute the differential:
\begin{align}f_{*,A}X=\frac{d}{dt}\Bigr|_{t=0}\bigr(f(\gamma(t)) \bigr)&=\frac{d}{dt}\Bigr|_{t=0}\Bigr( [\exp(tX^*A)A^*]\cdot[A\exp(tA^*X)]\Bigr)\\ &=X^*AA^*A+A^*AA^*X=X^*A+A^*X. \end{align} Here is where I am stuck, I do not know how to prove this is surjective.