#Study Notes: The Reciprocal of a Derivative and Derivatives of Inverse Functions

#Table of Contents

1. The Reciprocal of a Derivative
   • Concept Overview
   • Useful Applications
2. Derivatives of Inverse Functions
   • Relationship between Function and Inverse Derivatives
   • Deriving the Inverse Function Theorem
3. Worked Examples
4. Practice Questions
5. Glossary
6. Summary and Key Takeaways

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#The Reciprocal of a Derivative

#Concept Overview

Derivatives are not fractions, but they behave similarly when finding reciprocals. The reciprocal of a derivative is crucial in various calculations, especially involving inverse functions.

$$\frac{1}{\left( \frac{dy}{dx} \right)} = \frac{dx}{dy}$$

This holds true as long as $\frac{dy}{dx} \neq 0$.

Likewise, the reciprocal of $\frac{dx}{dy}$ is:

$$\frac{1}{\left( \frac{dx}{dy} \right)} = \frac{dy}{dx}$$

This is valid if $\frac{dx}{dy} \neq 0$.

#Useful Applications

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#Derivatives of Inverse Functions

#Relationship between Function and Inverse Derivatives

For a function $f$ that is differentiable and has an inverse $f^{-1}$, the derivative of the inverse at a point $a$ is given by:

$$\left( f^{-1} \right)'(a) = \frac{1}{f'\left( f^{-1}(a) \right)}$$

This is valid if $f'\left( f^{-1}(a) \right) \neq 0$.

You may also see this written as:

$$g'(a) = \frac{1}{f'\left( g(a) \right)}$$

Where $g(a) = f^{-1}(a)$.

This property is known as the Inverse Function Theorem.

#Deriving the Inverse Function Theorem

To derive the Inverse Function Theorem, we use the definition of an inverse and the chain rule:

1. Let $g(x) = f^{-1}(x)$.
2. Since $f$ and $g$ are inverses, we have $f(g(x)) = x$.
3. Differentiate both sides with respect to $x$:

$$\frac{d}{dx} \left( f(g(x)) \right) = \frac{d}{dx} (x)$$

4. Apply the chain rule to the left-hand side:

$$f'\left( g(x) \right) \cdot g'(x) = 1$$

5. Rearrange to solve for $g'(x)$:

$$g'(x) = \frac{1}{f'\left( g(x) \right)}$$

6. Recall that $g(x) = f^{-1}(x)$:

$$\left( f^{-1} \right)'(x) = \frac{1}{f'\left( f^{-1}(x) \right)}$$

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#Worked Examples

#Example 1

Given that $f(0) = 256$, find the value of $g'(256)$.

Write the inverse function theorem using $f(x)$ and $f^{-1}(x) = g(x)$:

$$g'(256) = \frac{1}{f'\left( g(256) \right)}$$

Since $f(0) = 256$, then $g(256) = 0$. Now find $f'(0)$:

$$f'(x) = 12(3x + 4)^3$$

Evaluate at $x = 0$:

$$f'(0) = 12(3(0) + 4)^3 = 12(4)^3 = 768$$

So,

$$g'(256) = \frac{1}{768}$$

#Example 2

Show that the inverse of $f(x) = x^3 + 2x - 10$ exists and find the derivative of the inverse at $x = 125$.

First, check if the inverse exists:

$$f'(x) = 3x^2 + 2$$

Since $f'(x)$ is always positive, $f(x)$ is one-to-one and thus $f^{-1}(x)$ exists.

Use the inverse function theorem:

$$\left( f^{-1} \right)'(125) = \frac{1}{f'\left( f^{-1}(125) \right)}$$

Find $f^{-1}(125)$. Solve:

$$125 = y^3 + 2y - 10$$

By solving, $y = 5$:

$$f^{-1}(125) = 5$$

Thus,

$$\left( f^{-1} \right)'(125) = \frac{1}{f'(5)}$$

Evaluate $f'(5)$:

$$f'(5) = 3(5)^2 + 2 = 77$$

So,

$$\left( f^{-1} \right)'(125) = \frac{1}{77}$$

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#Practice Questions

Practice Question

1. Given $f(x) = x^2 + 1$, find $\left( f^{-1} \right)'(2)$.

Practice Question

2. If $h(x) = \ln(x)$, find $\left( h^{-1} \right)'(e)$.

Practice Question

3. Show that the inverse of $f(x) = e^x + x$ exists and find $\left( f^{-1} \right)'(2)$.

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#Glossary

• Derivative: A measure of how a function changes as its input changes.
• Inverse Function: A function that "reverses" another function.
• Chain Rule: A formula for computing the derivative of the composition of two or more functions.
• Inverse Function Theorem: A theorem that provides a formula for the derivative of the inverse of a function.

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#Summary and Key Takeaways

#Summary

• The reciprocal of a derivative $\frac{dy}{dx}$ is $\frac{dx}{dy}$.
• The Inverse Function Theorem provides a method to find the derivative of the inverse function.
• The theorem is derived using the chain rule and the definition of an inverse function.

#Key Takeaways

• Understand the reciprocal relationship between $\frac{dy}{dx}$ and $\frac{dx}{dy}$.
• Apply the Inverse Function Theorem to find the derivative of inverse functions.
• Use the chain rule effectively in derivations and calculations.

Exam Tip

Remember, always verify if the function is one-to-one before applying the Inverse Function Theorem.

Common Mistake

A common mistake is to forget that the reciprocal relationship only holds when the derivative is non-zero.