# Calculus Examples

,

Step 1

To find elasticity of demand, use the formula .

Step 2

Step 2.1

Substitute for .

Step 2.2

Multiply by .

Step 2.3

Subtract from .

Step 3

Step 3.1

Rewrite the equation as .

Step 3.2

Subtract from both sides of the equation.

Step 3.3

Divide each term in by and simplify.

Step 3.3.1

Divide each term in by .

Step 3.3.2

Simplify the left side.

Step 3.3.2.1

Cancel the common factor of .

Step 3.3.2.1.1

Cancel the common factor.

Step 3.3.2.1.2

Divide by .

Step 3.3.3

Simplify the right side.

Step 3.3.3.1

Simplify each term.

Step 3.3.3.1.1

Move the negative in front of the fraction.

Step 3.3.3.1.2

Multiply by .

Step 3.3.3.1.3

Factor out of .

Step 3.3.3.1.4

Separate fractions.

Step 3.3.3.1.5

Divide by .

Step 3.3.3.1.6

Divide by .

Step 3.3.3.1.7

Multiply by .

Step 3.3.3.1.8

Divide by .

Step 4

Step 4.1

Differentiate the demand function.

Step 4.2

By the Sum Rule, the derivative of with respect to is .

Step 4.3

Evaluate .

Step 4.3.1

Since is constant with respect to , the derivative of with respect to is .

Step 4.3.2

Differentiate using the Power Rule which states that is where .

Step 4.3.3

Multiply by .

Step 4.4

Differentiate using the Constant Rule.

Step 4.4.1

Since is constant with respect to , the derivative of with respect to is .

Step 4.4.2

Add and .

Step 5

Step 5.1

Substitute for .

Step 5.2

Substitute the values of and .

Step 5.3

Cancel the common factor of and .

Step 5.3.1

Factor out of .

Step 5.3.2

Cancel the common factors.

Step 5.3.2.1

Factor out of .

Step 5.3.2.2

Cancel the common factor.

Step 5.3.2.3

Rewrite the expression.

Step 5.4

Combine and .

Step 5.5

Divide by .

Step 5.6

The absolute value is the distance between a number and zero. The distance between and is .

Step 6

Since , the demand is inelastic.