# Neustadt and Hadamard derivative

can anyone tell me what the difference is between the Hadamard and Neustadt derivatives? Cause it looks to me like they're very similar.

Hadamard derivative:

We say $$\phi:X\to Y$$ ($$X,Y$$ are Banach-Spaces) is Hadamard directionally differentiable at $$x\in X$$ in direction $$h\in X$$ if there exist a map $$A_x:X\to Y$$ such that \begin{align} \lim\limits_{t_n\to 0^+}\lim\limits_{h_n\to h}\frac{\phi(x+t_n h_n)-\phi(x)}{t_n}=A_x(h)\quad\forall h_n\to h,\quad\forall t_n\to 0^+ \end{align}

Neustadt derivative:

$$\psi:X\to Y$$ with $$\psi$$ continous. We say $$\psi$$ is Neustadt differntiable if for every $$x\in X$$ there exist a linear continous mapping $$B_{x}:X\to Y$$ such that \begin{align} \lim\limits_{t_n\to 0}\lim\limits_{x_n\to\tilde{x}} \frac{\psi(x+t_nx_n)-\psi(x)}{t_n}=B_{x}(\tilde{x}) \quad\forall \tilde{x}\in X \end{align}

• It would be nice if you could formulate these derivatives in your question. – MachineLearner Mar 6 at 9:06
• It looks that Neustadt implicate Hadamard. – FuncAna09 Mar 6 at 9:30