Inverse Function Theorem
The Inverse Function Theorem is a fundamental result in calculus that provides conditions under which a function has an inverse function that is also differentiable. If a function is continuously differentiable and its derivative is non-zero at a point, then the function has a differentiable inverse in a neighborhood of that point. The theorem also provides a formula for the derivative of the inverse function in terms of the original function’s derivative.
In particular, if a function f(x) is continuously differentiable and its derivative f'(x) is non-zero at a point x = a, then there exists an interval around x = a where f(x) has a differentiable inverse function, denoted by f-1(x). Moreover, the derivative of the inverse function is given by:
(f-1)'(x) = 1 / f'(f-1(x))