Problem Interpretation

We are tasked with finding the derivative of the inverse function, $(f^{-1}(x))'$, for the given function $f(x) = x^5 + x$, at $x = 2$.

Step-by-Step Solution:

1. Inverse Function Derivative Formula: The formula for the derivative of the inverse function is: $\frac{d}{dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}$

2. Find $f^{-1}(2)$: This requires solving $f(x) = 2$. The function $f(x) = x^5 + x$ is given: $f(x) = x^5 + x = 2$ Testing values: $f(1) = 1^5 + 1 = 2$ Thus, $f^{-1}(2) = 1$.

3. Compute $f'(x)$: The derivative of $f(x)$ is: $f'(x) = 5x^4 + 1$

4. Evaluate $f'(f^{-1}(2))$: Since $f^{-1}(2) = 1$, we substitute $x = 1$ into $f'(x)$: $f'(1) = 5(1)^4 + 1 = 5 + 1 = 6$

5. Find $(f^{-1}(x))'$ at $x = 2$: Using the formula: $\frac{d}{dx} f^{-1}(2) = \frac{1}{f'(f^{-1}(2))} = \frac{1}{6}$

Final Answer:

The value of $\frac{d}{dx} f^{-1}(x)$ at $x = 2$ is: $\boxed{\frac{1}{6}}$

Would you like additional details on this?
Here are some related questions to explore:

1. What is the general relationship between a function and its inverse regarding derivatives?
2. Can you derive the inverse function derivative formula step by step?
3. How do you verify if a function is invertible?
4. What happens if $f'(x) = 0$ at a point—how does this impact the inverse derivative?
5. Why is it necessary to solve $f(x) = 2$ in the context of this problem?

Tip:

When solving for the inverse derivative, always ensure the original function is one-to-one (invertible) in the relevant domain!