Behaviors of Implicit Relations

David Brown

5 min read

Next Topic - Critical Points of Implicit Relations

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Study Guide Overview

This study guide covers second derivatives of implicit functions. It starts with the fundamentals of implicit differentiation and proceeds to the steps for finding the second derivative. It includes a worked example, practice questions, and a glossary of terms like implicit differentiation, quotient rule, and chain rule. Finally, it offers a summary of key takeaways and exam strategies.

#Second Derivatives of Implicit Functions

#Introduction

Finding the second derivative of an implicit function involves differentiating the first derivative. This guide will walk you through the steps necessary to achieve this, using examples and providing tips to aid your understanding.

#Fundamentals of Implicit Differentiation

Before diving into second derivatives, ensure you're comfortable with implicit differentiation:

Implicit Differentiation involves finding the derivative of a function defined implicitly by using the chain rule.

#Finding the Second Derivative

To find the second derivative of an implicit function, follow these steps:

Find the First Derivative:
- Start with the implicit equation. For example, consider $x^2 + y^2 = 4x$ .
- Differentiate both sides with respect to $x$ .
$\frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(4x)$
- Appl...

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