# Calculus Examples

,
Step 1
To find elasticity of demand, use the formula .
Step 2
Substitute for in and simplify to find .
Step 2.1
Substitute for .
Step 2.2
Multiply by .
Step 2.3
Subtract from .
Step 3
Solve the demand function for .
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
Find by differentiating the demand function.
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
Step 5
Substitute into the formula for elasticity and simplify.
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.