# Orthogonal derivative implies second derivative is null

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.