partial derivatives of a harmonic function are harmonic

Question

Let $U \subset \Bbb R^n$ and $f: U \to \Bbb R$ be a harmonic function. I wanna show that the partial derivatives of a harmonic function are harmonic. So for $g= \frac{\partial f}{\partial x_j}$, I wanna show that $\sum_i \frac{\partial^2 g}{\partial x_i^2}=0$. ( $j \in \{1, \dots , n\}$) I know that f is a function of class $C^2$ but than?

Thanks for your help.