The $\sqrt{n}$ factors can be absorbed in the $X$ matrices, I can omit them. Consider a Hermitian matrix $X$ and a Hermitian perturbation $\delta X$. Then, for $f:\mathbb{R}\rightarrow\mathbb{R}$ one has $$\operatorname{Tr}f(X+\delta X)=\sum_{i}f(X)_{ii}+\sum_{i}f'(X)_{ii}\delta X_{ii}+\sum_{i<j}\left[f'(X)_{ji}\delta X_{ij}+\overline{f'(X)}_{ji}\overline{\delta X}_{ij}\right]+{\cal O}(\delta X_{ij})^2$$ $$\Rightarrow \frac{\partial}{\partial X_{ij}}\operatorname{Tr}f(X)=\begin{cases}f'(X)_{ii}&\text{if}\;\;i=j\\ f'(X)_{ji}&\text{if}\;\;i< j,\end{cases}$$ and also $$\frac{\partial}{\partial \overline{X}_{ij}}\operatorname{Tr}f(X)=\overline{f'(X)}_{ji}=f'(X)_{ij}.$$ So $\alpha,\beta,\gamma$ are all equal to 1.

The $\sqrt{n}$ factors can be absorbed in the $X$ matrices, I can omit them. Consider a Hermitian matrix $X$ and a Hermitian perturbation $\delta X$. Then, for $f:\mathbb{R}\rightarrow\mathbb{R}$ one has $$\operatorname{Tr}f(X+\delta X)=\sum_{i}f(X)_{ii}+\sum_{i}f'(X)_{ii}\delta X_{ii}+\sum_{i<j}\left[f'(X)_{ji}\delta X_{ij}+\overline{f'(X)}_{ji}\overline{\delta X}_{ij}\right]+{\cal O}(\delta X)^2$$ $$\Rightarrow \frac{\partial}{\partial X_{ij}}\operatorname{Tr}f(X)=\begin{cases}f'(X)_{ii}&\text{if}\;\;i=j\\ f'(X)_{ji}&\text{if}\;\;i< j,\end{cases}$$ and also $$\frac{\partial}{\partial \overline{X}_{ij}}\operatorname{Tr}f(X)=\overline{f'(X)}_{ji}=f'(X)_{ij}.$$ So $\alpha,\beta,\gamma$ are all equal to 1.

The $\sqrt{n}$ factors can be absorbed in the $X$ matrices, I can omit them. Then, $$\operatorname{Tr}f(X+\delta X)=\sum_{i}f(X)_{ii}+\sum_{i}f'(X)_{ii}\delta X_{ii}+2\sum_{i<j}f'(X)_{ij}\delta X_{ij}+{\cal O}(\delta X_{ij})^2$$ $$\Rightarrow \frac{\partial}{\partial X_{ij}}\operatorname{Tr}f(X)=\begin{cases}f'(X)_{ii}&\text{if}\;\;i=j\\ 2f'(X)_{ij}&\text{if}\;\;i\neq j.\end{cases}$$ So $\alpha=1$ and $\beta=2$.

The $\sqrt{n}$ factors can be absorbed in the $X$ matrices, I can omit them. Consider a Hermitian matrix $X$ and a Hermitian perturbation $\delta X$. Then, for $f:\mathbb{R}\rightarrow\mathbb{R}$ one has $$\operatorname{Tr}f(X+\delta X)=\sum_{i}f(X)_{ii}+\sum_{i}f'(X)_{ii}\delta X_{ii}+\sum_{i<j}\left[f'(X)_{ji}\delta X_{ij}+\overline{f'(X)}_{ji}\overline{\delta X}_{ij}\right]+{\cal O}(\delta X_{ij})^2$$ $$\Rightarrow \frac{\partial}{\partial X_{ij}}\operatorname{Tr}f(X)=\begin{cases}f'(X)_{ii}&\text{if}\;\;i=j\\ f'(X)_{ji}&\text{if}\;\;i< j,\end{cases}$$ and also $$\frac{\partial}{\partial \overline{X}_{ij}}\operatorname{Tr}f(X)=\overline{f'(X)}_{ji}=f'(X)_{ij}.$$ So $\alpha,\beta,\gamma$ are all equal to 1.