# 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* = *x*^{2} + 3

*y* = *x *cos *x*

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

For example:

*x*^{2} + *y*^{2} = 16

*x*^{2} + *y*^{2} = 4*xy*

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 *x*^{2} + * y*^{2} = 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 *x*^{3} + *y*^{3} = 6*xy*

**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