Related Topics: More Lessons on Calculus

In this lesson, we will learn how implicit differentiation can be used the find the derivatives of equations that are not functions.

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*

**Implicit Differentiation - Basic Idea and Examples**

What is implicit differentiation?

The basic idea about using implicit differentiation

1. Take derivative, adding dy/dx where needed

2. Get rid of parenthesis

3. Solve for dy/dx

Examples:

Find dy/dx.

x^{2} + xy + cos(y) = 8y
**Implicit Differentiation**

Examples

1. Find dy/dx

1 + x = sin(xy^{2})

2. Find the equation of the tangent line at (1, 1) on the curve x^{2} + xy + y^{2} = 3
**Examples of Implicit Differentiation**

1. x^{3} + y^{3} = xy

2. (x^{2}y) + (xy^{2}) = 3x
**How to use Implicit Differentiation to find a Derivative?**

Find the second derivative using implicit differentiation

Find y^{n} for:

9x^{2} + y^{2} = 9

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In this lesson, we will learn how implicit differentiation can be used the find the derivatives of equations that are not functions.

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

For example:

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'.

**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:**

What is implicit differentiation?

The basic idea about using implicit differentiation

1. Take derivative, adding dy/dx where needed

2. Get rid of parenthesis

3. Solve for dy/dx

Examples:

Find dy/dx.

x

Examples

1. Find dy/dx

1 + x = sin(xy

2. Find the equation of the tangent line at (1, 1) on the curve x

1. x

2. (x

Find the second derivative using implicit differentiation

Find y

9x

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You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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