# Implicit Differentiation

In this lesson, we will learn how implicit differentiation can be used the find the derivatives of equations that are not functions.
Related Topics: More Lessons on Calculus

Some functions can be described by expressing one variable explicitly in terms of another variable.

For example:
y = x2 + 3
y = x cos x

However, some equations are defined implicitly by a relation between x and y.

For example:
x2 + y2 = 16
x2 + y2 = 4xy

We do not need to solve an equation for y in terms of x in order to find the derivative of y. Instead, we can use the method of implicit differentiation. This involves differentiating both sides of the equation with respect to x and then solving the resulting equation for y’.

A set is a collection of objects, things or symbols which are clearly defined.
The individual objects in a set are called the members or elements of the set.
A set must be properly defined so that we can find out whether an object is a member of the set.

## 1. Listing the elements (Listing Method)

The set can be defined by listing all its elements, separated by commas and enclosed within braces. This is called the roster method.

Example:
B = {2, 4, 6, 8, 10}
X = {a, b, c, d, e}

However, in some instances, it may not be possible to list all the elements of a set. In such cases, we could define the set by method 2.

Example:

If x2 + y2 = 16, find

Solution:

Step 1: Differentiate both sides of the equation

Step 2: Using the Chain Rule, we find that

Step 3: Substitute equation (2) into equation (1)

Step 4: Solve for

Example:

Find y’ if x3 + y3 = 6xy

Solution:

## Videos

Implicit Differentiation - Basic Idea and Examples
The basic idea about using implicit differentiation

More! Implicit Differentiation Examples

More Implicit Differentiation Examples - 3

Using Implicit Differentiation to Find a Derivative
Using Implicit Differentiation to Find a Derivative