Problem 291 - Calculus Volume 1

Question

For the following exercises, use the given values to find \(\left(f^{-1}\right)^{\prime}(a).\) $$f\left(\frac{1}{3}\right)=-8, f^{\prime}\left(\frac{1}{3}\right)=2, a=-8$$

Step-by-Step Solution

Answer

\((f^{-1})^{\prime}(-8) = \frac{1}{2}.\)

Step 1: Understand the Given Information

We are given that \(f\left(\frac{1}{3}\right) = -8\), \(f'\left(\frac{1}{3}\right) = 2\), and \(a = -8\). We know \(f\) is invertible, and we need to find \((f^{-1})^{\prime}(a)\).

Step 2: Inverse Function Derivative Formula

The derivative of an inverse function is given by the formula \((f^{-1})^{\prime}(a) = \frac{1}{f'(f^{-1}(a))}\). Our task is to find \((f^{-1})^{\prime}(-8)\).

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

Since \(f(x) = -8\) when \(x = \frac{1}{3}\), it follows that \(f^{-1}(-8) = \frac{1}{3}\).

Step 4: Apply the Inverse Derivative Formula

We substitute \(\frac{1}{3}\) into the derivative formula: \[(f^{-1})^{\prime}(-8) = \frac{1}{f'(\frac{1}{3})}\].We know \(f'(\frac{1}{3}) = 2\). Therefore, \[(f^{-1})^{\prime}(-8) = \frac{1}{2}.\]

Key Concepts

Understanding Inverse Functions

Inverse functions play a pivotal role in calculus. Essentially, an inverse function reverses the operations of a given function. If you have a function \( f \), its inverse, denoted as \( f^{-1} \), operates such that if \( f(x) = y \), then \( f^{-1}(y) = x \). Here, the important part is that the function must be one-to-one (bijective), meaning it should be both injective (no two different inputs give the same output) and surjective (everything in the output has a corresponding input). This ensures that each output has exactly one corresponding input, making the inverse exist.
In the context of calculus, finding inverses often means solving for \( x \) in terms of \( y \). It's like swapping roles between input and output and is critical in finding derivative inverses as we saw in the exercise. Thus, grasping inverse functions lays the groundwork for more advanced operations like finding their derivatives.

Differentiating Inverse Functions

The derivative of an inverse function can be intriguing yet fascinating. The formula to find the derivative of an inverse function \( (f^{-1})^{\prime}(a) \) is fundamental and requires understanding of both the function and its derivative. Mathematically, we express this as:
\[(f^{-1})^{\prime}(a) = \frac{1}{f'(f^{-1}(a))}\]
This equation tells us how the rate of change of the inverse function can be expressed using the original function's derivative. The key step is to correctly identify \( f^{-1}(a) \), which means finding the input \( x \) that gives the output \( a \) for the function \( f \). Once known, simply substitute this value into the derivative of the original function and follow the formula.
Being able to differentiate inverse functions is valuable, especially when dealing with complex functions where explicit inverses are challenging to find.

Approaching Calculus Problem Solving

Calibrating your approach to calculus problems can greatly influence your efficiency and accuracy. The steps demonstrated in handling the inverse function derivative problem highlight a clear methodology:

• Comprehend the Problem: Understand what is being asked and deciphering given data.
• Utilize Correct Formulas: Identify and apply the right mathematical formulas and theorems relevant to the problem. In this case, the inverse function derivative formula.
• Solve Methodically: Work through the calculations step by step, ensuring all substitutions and mathematical operations are correctly executed.

Such structured problem-solving not only aids in arriving at the correct solution but also enhances overall mathematical intuition. Remember, practice makes perfect, especially with calculus. Constantly tackle diverse problem sets to get better acquainted with various approaches.