Problem 1 - Calculus Single Variable

Question

In Exercises \(1-8,\) assume that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is invertible and differentiable. Compute \(\left(f^{-1}\right)^{\prime}(4)\) from the given information. $$ f^{-1}(4)=1, f^{\prime}(1)=2 $$

Step-by-Step Solution

Answer

\( \left(f^{-1}\right)^{\prime}(4) = \frac{1}{2} \)

Step 1: Understanding the Relationship

We are given that the function \( f \) is invertible and differentiable. We know \( f^{-1}(4) = 1 \), which implies \( f(1) = 4 \). This indicates \( 1 \) is the input for \( f \) that gives us the output \( 4 \). Also, we are given \( f'(1) = 2 \). Our goal is to find the derivative of the inverse function at the point where its output is \( 4 \).

Step 2: Utilizing the Inverse Function Derivative Formula

The formula for the derivative of an inverse function is given by: \[ \left( f^{-1} \right)'(b) = \frac{1}{f'(a)}, \] where \( a = f^{-1}(b) \). In this particular exercise, \( b = 4 \) and therefore \( a = f^{-1}(4) = 1 \). Since we know that \( f'(1) = 2 \), we can substitute these values into the formula.

Step 3: Calculating the Derivative of the Inverse

Substitute \( a=1 \) and the given derivative value \( f'(1) = 2 \) into the inverse function derivative formula: \[(f^{-1})'(4) = \frac{1}{f'(1)} = \frac{1}{2}.\] This gives us the derivative of the inverse function at the point where its output is \( 4 \).

Key Concepts

Invertible Functions

In mathematics, an invertible function is one that has a unique inverse. This essentially means that every output from the function can be traced back to a single input. Such functions are commonly referred to as "one-to-one." A function being one-to-one is critical for it to be invertible. Otherwise, the concept of an inverse wouldn't stand as each output must link directly to just one input.

For example, if a function \( f \) maps every number in a particular way, then an inverse function \( f^{-1} \) will map back from every output to its original input. It's this symmetric property that defines inversible functions and makes them important in various mathematical analyses:

• Ensures one-to-one correspondence between inputs and outputs.
• Allows for the computation of original inputs given the output using the inverse function.

When dealing with inverse functions, structure and consistency are key. This also facilitates computations involving their derivatives, such as discovering how changes in input reflect changes in output.

Differentiable Functions

Differentiable functions are those for which a tangent line or derivative function can be defined at every point in their domain. This means that the function behaves smoothly and has no abrupt changes or "jumps" in its curve. Differentiability is a fundamental concept in calculus, as it allows us to discuss rates of change and slopes.

For a function to be differentiable:

• It must be continuous—no breaks or gaps in the graph.
• It must have a defined slope at every point—no sharp corners or cusps that would break differentiability.

Differentiability and continuity go hand-in-hand, yet not every continuous function is differentiable at every point. It's critical to grasp this for solving problems related to the derivatives of functions and their inverses, as being able to calculate these derivatives hinges on the function being differentiable at the requisite points.

Chain Rule

The chain rule is a fundamental formula in calculus for differentiating composite functions. These are functions composed of one function within another, symbolized as \( g(f(x)) \). It tells us how to differentiate such compositions by handling each function separately and then combining their derivatives.

The chain rule formula is expressed as:
\[ (g \circ f) '(x) = g'(f(x)) \cdot f'(x) \]

This critical calculus tool allows you to break down complex functions and effectively solve for their derivatives by:

• First differentiating the outer function \( g(x) \).
• Evaluating at the inner function \( f(x) \).
• Multiplying by the derivative of the inner function \( f'(x) \).

Understanding and applying the chain rule is vital in dealing with more complicated derivatives, such as when you have to compute the derivative of inverse functions or when functions are nested inside each other. With mastery of the chain rule, any multi-layered problem becomes manageable, providing a pathway through the layers of functions.