Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>
b. (f^-1)'(6)

Verified step by step guidance

1. Identify the relationship between the function \( f \) and its inverse \( f^{-1} \). Recall that if \( y = f(x) \), then \( x = f^{-1}(y) \).
2. Use the formula for the derivative of an inverse function: \((f^{-1})'(b) = \frac{1}{f'(a)}\), where \( f(a) = b \).
3. From the problem, we need to find \((f^{-1})'(6)\). This means we need to find \( a \) such that \( f(a) = 6 \).
4. Look at the table provided to find the value of \( a \) for which \( f(a) = 6 \).
5. Once \( a \) is identified, find \( f'(a) \) from the table and use the formula \((f^{-1})'(6) = \frac{1}{f'(a)}\) to determine the derivative.

Key Concepts

Inverse Functions

An inverse function reverses the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f^-1(y) takes y back to x. Understanding how to find and interpret inverse functions is crucial for solving problems involving derivatives of these functions.

Derivative of Inverse Functions

The derivative of an inverse function can be calculated using the formula (f^-1)'(y) = 1 / f'(x), where y = f(x). This relationship shows that the rate of change of the inverse function at a point is the reciprocal of the rate of change of the original function at the corresponding point. This concept is essential for determining the derivative of f^-1 at a specific value.

Using Tables for Derivatives

In calculus, tables can provide values of functions and their derivatives at specific points. When asked to find the derivative of an inverse function using a table, one must locate the corresponding values for f and f' to apply the inverse derivative formula. This method is particularly useful when explicit functions are not available.