Problem 290 - Calculus Volume 1

Question

Use the given values to find \(\left(f^{-1}\right)^{\prime}(a)\). $$ f(6)=2, f^{\prime}(6)=\frac{1}{3}, a=2 $$

Step-by-Step Solution

Answer

\( \left(f^{-1}\right)^{\prime}(2) = 3 \).

Step 1: Understand the Problem

We need to find the derivative of the inverse function at a given point, specifically \( \left(f^{-1}\right)^{\prime}(a) \). We're given values \( f(6)=2 \), \( f^{\prime}(6)=\frac{1}{3} \), and \( a=2 \).

Step 2: Apply Inverse Function Derivative Rule

The rule for the derivative of an inverse function states: if \( f(x) \) is differentiable and its inverse \( f^{-1}(x) \) exists and is differentiable at \( a \), then \( \left(f^{-1}\right)^{\prime}(a) = \frac{1}{f^{\prime}(f^{-1}(a))} \). We need to find \( f^{-1}(2) \) first.

Step 3: Determine \( f^{-1}(a) \)

Given \( f(6) = 2 \), this implies that \( f^{-1}(2) = 6 \). Because applying \( f \) to 6 returns 2, \( f^{-1}(2) \) must return 6.

Step 4: Substitute Values into Derivative Rule

We substitute \( f^{-1}(2) = 6 \) into the derivative formula: \( \left(f^{-1}\right)^{\prime}(2) = \frac{1}{f^{\prime}(6)} \).

Step 5: Calculate the Final Answer

Using the given value \( f^{\prime}(6) = \frac{1}{3} \), substitute to find \( \left(f^{-1}\right)^{\prime}(2) = \frac{1}{\frac{1}{3}} = 3 \).

Key Concepts

Derivative of Inverse Functions

The derivative of inverse functions is a pivotal concept in calculus. When you have a function and its inverse, you might wonder how their derivatives relate to one another. The rule for the derivative of an inverse function provides a neat solution. If a function \( f(x) \) is continuous and differentiable, and its inverse \( f^{-1}(x) \) exists, the relationship between their derivatives can be expressed as:

• \( \left( f^{-1} \right)^{\prime}(a) = \frac{1}{f^{\prime}(f^{-1}(a))} \).

This formula allows us to find the derivative of the inverse function at a specific point \( a \). It's crucial to first determine the inverse's value at \( a \) before applying the formula. This relationship simplifies the task of finding derivatives of inverse functions without directly inverting the function itself.

Inverse Function

An inverse function essentially "undoes" what the original function does. If function \( f \) maps a value \( x \) to \( y \), the inverse function \( f^{-1} \) maps \( y \) back to \( x \). Understanding this concept is essential when dealing with derivative problems involving inverse functions.

• The inverse exists if, for every output of the original function, there's exactly one corresponding input.
• The role of the inverse function is to reverse the effect of the function.

For example, if \( f(6) = 2 \), then \( f^{-1}(2) = 6 \). This relationship is vital in applying the derivative of inverse function rule in calculus.

Calculus Problem-Solving

Solving calculus problems requires a systematic approach. For the exercise at hand, we're tasked with finding the derivative of the inverse function at a given point. Here's a simplified approach:

• Identify the given values and what is required. In this case, \( f(6) = 2 \), \( f^{\prime}(6) = \frac{1}{3} \), and \( a = 2 \).
• Determine the inverse value, such as \( f^{-1}(2) = 6 \) since \( f(6) = 2 \).
• Apply the derivative of inverse function formula: \( \left( f^{-1} \right)^{\prime}(a) = \frac{1}{f^{\prime}(f^{-1}(a))} \).
• Plug in the known values and compute the result: \( \left( f^{-1} \right)^{\prime}(2) = \frac{1}{\frac{1}{3}} = 3 \).

By following these steps, you can systematically solve derivatives involving inverse functions, ensuring accuracy and clarity in your solutions.

Function Derivatives

Derivatives are a fundamental concept in calculus, representing the rate of change or slope of a function at any given point. When dealing with function derivatives, especially with inverse functions, there are certain nuances to keep in mind.

• The derivative of a function \( f \) at a point is denoted as \( f^{\prime}(x) \).
• Knowing the derivatives of base functions can help in calculating more complex derivatives.

In our exercise, knowing \( f^{\prime}(6) = \frac{1}{3} \) was key. This allowed us to determine the derivative of the inverse, showcasing how interconnected and essential understanding function derivatives is, whether for regular or inverse functions.