Suppose that the differentiable function y = f(x) has an inverse and that the graph of f passes through the point (2, 4) and has a slope of 1/3 there. Find the value of df⁻¹/dx at x = 4.

검증된 단계별 안내

Recall that if \( y = f(x) \) has an inverse function \( f^{-1}(x) \), then the derivative of the inverse at a point \( x \) is given by the formula: \[ \frac{d}{dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(x))} \]

Identify the point on the original function \( f \) given in the problem: the function passes through \( (2, 4) \) and has a slope \( f'(2) = \frac{1}{3} \).

Since \( f(2) = 4 \), the inverse function satisfies \( f^{-1}(4) = 2 \). This means when \( x = 4 \) in the inverse function, the corresponding \( y \) value is 2.

Apply the formula for the derivative of the inverse at \( x = 4 \): \[ \frac{d}{dx} f^{-1}(4) = \frac{1}{f'(f^{-1}(4))} = \frac{1}{f'(2)} \]

Substitute the known slope \( f'(2) = \frac{1}{3} \) into the expression to find \( \frac{d}{dx} f^{-1}(4) \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. If y = f(x) is invertible, then f⁻¹(y) returns the original x. Understanding this relationship is crucial for connecting values of f and f⁻¹ at corresponding points.

Derivative of an Inverse Function

The derivative of the inverse function at a point x is the reciprocal of the derivative of the original function at the corresponding point y = f⁻¹(x). Formally, (df⁻¹/dx)(x) = 1 / (df/dx)(f⁻¹(x)). This formula allows us to find the slope of the inverse function using the slope of the original function.

Evaluating Derivatives at Specific Points

To find the derivative of the inverse at x = 4, identify the corresponding point on f where f(x) = 4. Using the given point (2, 4), we know f(2) = 4, so f⁻¹(4) = 2. Then, use the slope of f at x = 2 to compute the derivative of the inverse at x = 4.