Find (f^−1)′(3), where f(x)=x³+x+1.

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the derivative of the inverse function \( (f^{-1})'(3) \) for the function \( f(x) = x^3 + x + 1 \).
2. Step 2: Use the formula for the derivative of an inverse function: \( (f^{-1})'(y) = \frac{1}{f'(x)} \) where \( f(x) = y \).
3. Step 3: Find \( x \) such that \( f(x) = 3 \). This means solving the equation \( x^3 + x + 1 = 3 \) to find the value of \( x \).
4. Step 4: Compute the derivative \( f'(x) \) of the function \( f(x) = x^3 + x + 1 \). The derivative is \( f'(x) = 3x^2 + 1 \).
5. Step 5: Evaluate \( f'(x) \) at the \( x \) found in Step 3, and use the formula from Step 2 to find \( (f^{-1})'(3) = \frac{1}{f'(x)} \).