Frechet derivative

The Frechet derivative of $f$ at $a$ is the unique linear map $D{f}_a$ such that

A map on a subset of a normed vector space is Hadamard differentiable at $s$ if its directional difference quotients converge to a continuous linear map uniformly along all convergent sequences of directions: $t_j\to 0$, $a_j\to a$, and $s+t_j{a}_j$ in the domain imply $(\Phi (s+t_j{a}_j)-\Phi (s))/t_j\to {\dot{\Phi }}_s(a)$. Tangential differentiability restricts the limit directions to a specified subspace. This is the regularity used by the functional delta method.

On bounded continuously differentiable functions with uniformly bounded total variation of a function, the map $T$ is Hadamard differentiable in the supremum norm with derivative ${\dot{T}}_{F,G}(a,b)={\int }_0^{u}b{F}^{\prime }+G(u)a(u)-G(0)a(0)-{\int }_0^u{G}^{\prime }a$. Use weak convergence of bounded-variation integrators for the first term and integration by parts for the remaining terms. No derivatives of the limiting continuous directions are required.

A map is Fréchet differentiable at $x$ if it admits a bounded linear first-order approximation: $f(x+h)=f(x)+Ah+o(\Vert h\Vert )$ for a continuous linear map $A$.

On the open set of invertible square matrices, the inversion map is differentiable and