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}

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

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