Find (f^−1)′(3), where f(x)=x³+x+1.

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Step 1: Understand the problem. We need to find the derivative of the inverse function \( (f^{-1})'(3) \) for the function \( f(x) = x^3 + x + 1 \).

Step 2: Use the formula for the derivative of an inverse function: \( (f^{-1})'(y) = \frac{1}{f'(x)} \) where \( f(x) = y \).

Step 3: Find \( x \) such that \( f(x) = 3 \). This means solving the equation \( x^3 + x + 1 = 3 \) to find the value of \( x \).

Step 4: Compute the derivative \( f'(x) \) of the function \( f(x) = x^3 + x + 1 \). The derivative is \( f'(x) = 3x^2 + 1 \).

Step 5: Evaluate \( f'(x) \) at the \( x \) found in Step 3, and use the formula from Step 2 to find \( (f^{-1})'(3) = \frac{1}{f'(x)} \).

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주요 개념

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

Inverse Function

An inverse function reverses the effect of the original function. For a function f(x), its inverse f^(-1)(y) satisfies the equation f(f^(-1)(y)) = y. To find the derivative of an inverse function at a specific point, we often use the relationship between the derivatives of the original and inverse functions.

Derivative of Inverse Function

The derivative of an inverse function can be calculated using the formula (f^(-1))'(y) = 1 / f'(x), where y = f(x). This means that to find the derivative of the inverse at a point, we first need to determine the corresponding x-value such that f(x) equals the given y-value, and then compute the derivative of f at that x.

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative is given by dy/dx = f'(g(x)) * g'(x). This rule is essential when dealing with inverse functions, as it helps in understanding how changes in x affect y through the composition of functions.