This guide covers implicit functions and implicit differentiation. It explains how to differentiate equations involving both x and y using the chain rule, product rule, and quotient rule. It includes examples of finding derivatives at specific points and differentiating inverse functions implicitly. Practice questions and a glossary are also provided.

#Derivatives of Implicit Functions

#Introduction to Implicit Functions

#What is an Implicit Function?

An equation in the form $y=f(x)$ or $x=f(y)$ is written explicitly.

Exam Tip

E.g. $y=3x^2 + 2x - 3$

Equations involving both $x$ and $y$ are referred to as implicit functions.

Exam Tip

E.g. $3x^2 - 7xy^2 = 3$ or $x^2 + y^2 = 25$

For such equations, we cannot express $y$ solely in terms of $x$ or vice versa.
However, these equations define a relationship between $x$ and $y$ .

#Implicit Differentiation

#What is Implicit Differentiation?

Implicit differentiation is the method used to differentiate implicit functions.
To differentiate an implicit function with respect to $x$ , each term is differentiated with respect to $x$ .
For terms involving only $x$ , this is straightforward.
For terms involving $y$ , we apply the chain rule.

Key Concept

$\frac{d}{dx}f(y) = f'(y) \cdot \frac{dy}{dx}$ This means:

Differentiate the function in terms of $y$ with respect to $y$ .
Multiply it by $\frac{dy}{dx}$ .

Once each term has been differentiated, rearrange the equation to solve for $\frac{dy}{dx}$ .
Factorize out $\frac{dy}{dx}$ if necessary.
Substitute specific $(x, y)$ values to find the derivative at a point if required.

#How to Use Implicit Differentiation?

#Example

Differentiate $x^2 + y^2 =...

#Derivatives of Implicit Functions

#Table of Contents

Introduction to Implicit Functions
Implicit Differentiation
Examples and Applications
Differentiating Inverse Functions
Practice Questions
Glossary
Summary and Key Takeaways

#Introduction to Implicit Functions

#What is an Implicit Function?

An equation in the form $y=f(x)$ or $x=f(y)$ is written explicitly.

Exam Tip

E.g. $y=3x^2 + 2x - 3$

Equations involving both $x$ and $y$ are referred to as implicit functions.

Exam Tip

E.g. $3x^2 - 7xy^2 = 3$ or $x^2 + y^2 = 25$

For such equations, we cannot express $y$ solely in terms of $x$ or vice versa.
However, these equations define a relationship between $x$ and $y$ .

#Implicit Differentiation

#What is Implicit Differentiation?

Implicit differentiation is the method used to differentiate implicit functions.
To differentiate an implicit function with respect to $x$ , each term is differentiated with respect to $x$ .
For terms involving only $x$ , this is straightforward.
For terms involving $y$ , we apply the chain rule.

Key Concept

$\frac{d}{dx}f(y) = f'(y) \cdot \frac{dy}{dx}$ This means:

Differentiate the function in terms of $y$ with respect to $y$ .
Multiply it by $\frac{dy}{dx}$ .

Once each term has been differentiated, rearrange the equation to solve for $\frac{dy}{dx}$ .
Factorize out $\frac{dy}{dx}$ if necessary.
Substitute specific $(x, y)$ values to find the derivative at a point if required.

#How to Use Implicit Differentiation?

#Example

Differentiate $x^2 + y^2 =...

Which of the following equations represents an implicit function? 🤔

Implicit Differentiation

David Brown

#Derivatives of Implicit Functions

#Table of Contents

#Introduction to Implicit Functions

#What is an Implicit Function?

#Implicit Differentiation

#What is Implicit Differentiation?

#How to Use Implicit Differentiation?

#Example

How are we doing?

Implicit Differentiation

David Brown

#Derivatives of Implicit Functions

#Table of Contents

#Introduction to Implicit Functions

#What is an Implicit Function?

#Implicit Differentiation

#What is Implicit Differentiation?

#How to Use Implicit Differentiation?

#Example

How are we doing?