What kind of derivative is this? In this question here the equation of the derivative of the projection map $\pi_i : \mathbb R^n \to \mathbb R$ is given by
$$\pi_i(X+H)-\pi_i(X)=\textrm{grad}\ \pi_i(X)\cdot H+||H||g(H)$$

What definition of derivative is used here?

I thought it should be the Frechet derivative but it seems not.

That's the ordinary derivative; the equation you quoted is Taylor's theorem with remainder, written in a slightly unusual way:
$$f(x+h) = f(x) + h^T\nabla f(x) + ||h||g(x+h)$$
and $\lim_{h\to 0} g(x+h) = 0$.
