🔄 Inverse Function Derivative Calculator

Using the Inverse Function Theorem

To find the derivative of an inverse function, $f^{-1}'(a)$, without actually finding the inverse function $f^{-1}(x)$, we use the Inverse Function Theorem.

The Inverse Function Theorem

The theorem states that if $f$ is a differentiable function with an inverse $f^{-1}$, then the derivative of the inverse function at a point $a$ is the reciprocal of the derivative of the original function evaluated at the point $b$, where $f(b) = a$.

This powerful theorem simplifies the process into three main steps:

1. Find $b$: Solve the equation $f(b) = a$ for $b$. This $b$ value is equal to $f^{-1}(a)$.
2. Find $f'(x)$: Calculate the derivative of the original function, $f'(x)$.
3. Calculate Reciprocal: Evaluate $f'(b)$ and take its reciprocal. This gives you the final answer, $f^{-1}'(a)$.

Our calculator performs all these steps automatically, handling complex differentiation and numerical solving for the intermediate value $b$.

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