Orthogonal derivative implies second derivative is null

Question

Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be twice differentiable, such that $f'(x)$ is an orthogonal linear transformation for every $x\in\mathbb{R}^n$. Prove that $f''(x) = 0$, for every $x\in\mathbb{R}^n$. I've been struggling with this one, apparently it doesn't even use the inverse function theorem. Mi idea was to try and use the fact that $f'(x)$ preserves the norm, but that didn't work.