 Derivative of a local inverse map

For the continuously differentiable inverse function $g$ supplied by the inverse function theorem, the chain rule applied to $f(g(y))=y$ givesHere $I$ 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 $g^{\prime }(y)=1/f^{\prime }(g(y))$.

The formula also describes sensitivity. For a small output perturbation $h$, differentiability gives $g(y+h)-g(y)=Dg(y)h+o(\Vert h\Vert )$. A large operator norm of $Dg(y)$ permits large amplification in some directions. In the square root example, $g^{\prime }(y)=1/(2\sqrt{y})$ 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.