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

**Related Pages**

Calculus: Derivatives

Calculus: Derivative Rules

Calculus Lessons

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'.

**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:**

**Implicit Differentiation - Basic Idea and Examples**

What is implicit differentiation?

The basic idea about using implicit differentiation

- Take derivative, adding dy/dx where needed.
- Get rid of parenthesis.
- Solve for dy/dx

**Examples:**

Find dy/dx.

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

**Implicit Differentiation**

**Examples:**

- Find dy/dx

1 + x = sin(xy^{2}) - Find the equation of the tangent line at (1, 1) on the curve x
^{2}+ xy + y^{2}= 3

**Examples of Implicit Differentiation**

- x
^{3}+ y^{3}= xy - (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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