Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>
(g^-1)'(7)

Verified step by step guidance

1. Identify that you need to find the derivative of the inverse function, \((g^{-1})'(7)\).
2. Recall the formula for the derivative of an inverse function: \((f^{-1})'(b) = \frac{1}{f'(a)}\), where \(f(a) = b\).
3. Determine the value of \(a\) such that \(g(a) = 7\) using the table provided.
4. Once \(a\) is found, use the table to find \(g'(a)\), the derivative of \(g\) at \(a\).
5. Apply the formula \((g^{-1})'(7) = \frac{1}{g'(a)}\) to find the desired derivative value.

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the graph of the function at a given point. Derivatives can be computed using various rules and can also be interpreted as the limit of the average rate of change as the interval approaches zero.

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function g takes an input x and produces an output y, then its inverse g^-1 takes y back to x. The derivative of an inverse function can be found using the relationship (g^-1)'(y) = 1 / g'(x), where g(x) = y, highlighting the connection between the derivatives of a function and its inverse.

Using Tables for Derivatives

When derivatives are provided in a table format, it allows for quick reference to the values of a function and its derivatives at specific points. To find the derivative of an inverse function at a certain value, one must locate the corresponding value in the table for the original function, ensuring that the correct derivative is used in the calculation. This method is particularly useful when explicit functions are complex or not easily differentiable.