Find a formula for (f^-1)^'(x) given that f is one-to-one...

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Key Concepts

Inverse Function Theorem

This theorem provides the conditions under which a function has a local inverse. It asserts that if a function is continuously differentiable and its derivative is non-zero at a point, then locally, the function is invertible and its inverse is also differentiable. The theorem is crucial for establishing the existence of an inverse function and for calculating the derivative of the inverse function.

Derivative of the Inverse Function Formula

When a function f is invertible and differentiable, its inverse function f?¹ is also differentiable wherever f'(x) is non-zero. The derivative of the inverse function is given by (f?¹)'(y) = 1 / f'(f?¹(y)). This formula directly relates the rates of change of the function and its inverse, allowing one to compute the derivative of the inverse without finding the inverse function explicitly.

Differential Equations and Exponential Functions

A function that satisfies a differential equation of the form f'(x) = f(x) typically exhibits exponential growth or decay, leading to a general solution of the form f(x) = Ce^x. Recognizing this relationship helps in expressing the derivative of the function in terms of the function itself, which simplifies the computation of the derivative of its inverse using the formula from the inverse function theorem.

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