Let y = f(x) and f is invertible; then (f^{-1})'(y) equals what?

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Multiple Choice

Let y = f(x) and f is invertible; then (f^{-1})'(y) equals what?

Explanation:
Derivative of the inverse function is the key idea here. If y = f(x) and f is invertible and differentiable, then x = f^{-1}(y). Using the fact that f(f^{-1}(y)) = y and differentiating both sides with respect to y gives f'(f^{-1}(y)) · (f^{-1})'(y) = 1. Solving for the derivative of the inverse, (f^{-1})'(y) = 1 / f'(f^{-1}(y)). This requires f' to be nonzero at the corresponding x.

Derivative of the inverse function is the key idea here. If y = f(x) and f is invertible and differentiable, then x = f^{-1}(y). Using the fact that f(f^{-1}(y)) = y and differentiating both sides with respect to y gives f'(f^{-1}(y)) · (f^{-1})'(y) = 1. Solving for the derivative of the inverse, (f^{-1})'(y) = 1 / f'(f^{-1}(y)). This requires f' to be nonzero at the corresponding x.

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