= Derivative of a local inverse map
{title2=$Dg(y)=[Df(g(y))]^{-1}$}
For the <continuously differentiable> <inverse function> $g$ supplied by the <inverse function theorem>, the <chain rule> applied to $f(g(y))=y$ gives
$$
Df(g(y))Dg(y)=I,\qquad
Dg(y)=[Df(g(y))]^{-1}.
$$
Here $I$ is the <identity matrix>. The <matrix inverse> is taken at the recovered input $g(y)$, not at the output $y$. In one dimension this is the <inverse function rule> $g'(y)=1/f'(g(y))$.
The formula also describes sensitivity. For a small output perturbation $h$, <differentiability> gives $g(y+h)-g(y)=Dg(y)h+o(\|h\|)$. A large <operator norm> of $Dg(y)$ permits large amplification in some directions. In the <square root> example, $g'(y)=1/(2\sqrt y)$ grows without bound as positive $y$ approaches zero. Existence of a local <inverse function> therefore does not imply that recovery is insensitive to measurement error.
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