Inverse Function Theorem
(f^-1)'(a) = 1/f'(f^-1(a))
The Inverse Function Theorem is a theorem in calculus that relates to the properties of invertible functions. The theorem is a statement about the relationships between the derivatives of a function and its inverse. Specifically, the theorem states that if a function f is continuously differentiable in an open subset of the real numbers, and its derivative is nonzero at a point in that subset, then the function has an inverse function that is also continuously differentiable in an open subset of the real numbers.
More precisely, the Inverse Function Theorem is formulated as follows:
Let f : U → R^n be a continuously differentiable function defined on an open subset U of R^n. Suppose there exists a point p ∈ U such that the derivative matrix Df(p) is invertible. Then there exist neighborhoods U’ of p and V’ of f(p) such that:
1. f : U’ → V’ is a bijection from U’ onto V’.
2. The inverse function f^{-1} : V’ → U’ is continuously differentiable.
3. The derivative matrix Df^{-1}(q) is the inverse of the derivative matrix Df(p) for every q ∈ V’.
In other words, the Inverse Function Theorem guarantees that if a function has a nonzero derivative at a point, then it is locally invertible near that point, and its inverse function has the same smoothness properties as the original function.
The Inverse Function Theorem has many important applications in mathematics, physics, and engineering, especially in the study of differential equations, optimization theory, and control theory. It is also a key tool in the proofs of many fundamental theorems in calculus and analysis, such as the Implicit Function Theorem, the Global Inverse Function Theorem, and the Rank-Nullity Theorem.
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