The inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: its derivative is continuous and non-zero at the point.
The theorem also gives a formula for the derivative of the inverse function: In single variable calculus, the inverse rule states that .
In Multi-variate Context
A function has an inverse if and only if and for all and .
Computing inverse of a non-linear multi-variate functions is difficult. However, we can find a local inverse via Taylor expansion and linearization.
finding nonlinear inverses
Given
It is hard to find an inverse to this function. However, if we want to find an inverse to a particular point, say for close to zero, we can Taylor expand the function.
We will ignore the higher-order terms, and as a result we get
Now we get an linear function and the derivative is invertible.
The big idea: the derivative at a point tells you about the local nature of the function there, including invertibility.
The inverse rule in multi-variate context
The derivative of the inverse is the inverse of the derivative
assuming the function is differentiable
Proof
By definition and vice versa (via chain rule)
thus
Note that the existence of an inverse should never be taken for granted.
Inverse function theorem
is locally invertible near if the derivative of at is invertible