# Calculus Examples

Step 1
Find the first derivative.
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Differentiate using the Constant Rule.
Since is constant with respect to , the derivative of with respect to is .
Step 2
Set the first derivative equal to and solve for .
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
Simplify the numerator.
Raise to the power of .
Multiply .
Multiply by .
Multiply by .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply .
Multiply by .
Multiply by .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Change the to .
Rewrite as .
Factor out of .
Factor out of .
Move the negative in front of the fraction.
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply .
Multiply by .
Multiply by .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Change the to .
Rewrite as .
Factor out of .
Factor out of .
Move the negative in front of the fraction.
The final answer is the combination of both solutions.
Step 3
Split into separate intervals around the values that make the first derivative or undefined.
Step 4
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raise to the power of .
Multiply by .
Multiply by .
Simplify by subtracting numbers.
Subtract from .
Subtract from .
Step 5
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raising to any positive power yields .
Multiply by .
Multiply by .
Subtract from .
Step 6
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Multiply by by adding the exponents.
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Raise to the power of .
Multiply by .
Subtract from .
Step 7
Since the first derivative changed signs from positive to negative around , then there is a turning point at .
Step 8
Find the y-coordinate of to find the turning point.
Find to find the y-coordinate of .
Replace the variable with in the expression.
Simplify .
Remove parentheses.
Simplify each term.
Use the power rule to distribute the exponent.
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Raise to the power of .
Use the Binomial Theorem.
Simplify each term.
Raise to the power of .
Multiply by by adding the exponents.
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Raise to the power of .
Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Evaluate the exponent.
Multiply .
Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Rewrite as .
Raise to the power of .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Cancel the common factor of and .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Use the power rule to distribute the exponent.
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Multiply by .
Raise to the power of .
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Rewrite as .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Multiply .
Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Evaluate the exponent.
Multiply by .
Cancel the common factor of and .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply .
Multiply by .
Combine and .
To write as a fraction with a common denominator, multiply by .
Combine fractions.
Combine and .
Simplify the expression.
Combine the numerators over the common denominator.
Multiply by .
Simplify the numerator.
Apply the distributive property.
Multiply by .
Multiply by .
Find the common denominator.
Write as a fraction with denominator .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Write as a fraction with denominator .
Multiply by .
Multiply by .
Multiply by .
Combine the numerators over the common denominator.
Simplify each term.
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Multiply by .
Subtract from .
Write the and coordinates in point form.
Step 9
Since the first derivative changed signs from negative to positive around , then there is a turning point at .
Step 10
Find the y-coordinate of to find the turning point.
Find to find the y-coordinate of .
Replace the variable with in the expression.
Simplify .
Remove parentheses.
Simplify each term.
Use the power rule to distribute the exponent.
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Raise to the power of .
Use the Binomial Theorem.
Simplify each term.
Raise to the power of .
Multiply by by adding the exponents.
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Raise to the power of .
Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Evaluate the exponent.
Multiply .
Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Rewrite as .
Raise to the power of .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Subtract from .
Cancel the common factor of and .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Use the power rule to distribute the exponent.
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Multiply by .
Raise to the power of .
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Rewrite as .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Multiply .
Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Evaluate the exponent.
Multiply by .
Subtract from .
Cancel the common factor of and .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply .
Multiply by .
Combine and .
To write as a fraction with a common denominator, multiply by .
Combine fractions.
Combine and .
Simplify the expression.
Combine the numerators over the common denominator.
Multiply by .
Simplify the numerator.
Apply the distributive property.
Multiply by .
Multiply by .
Find the common denominator.
Write as a fraction with denominator .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Write as a fraction with denominator .
Multiply by .
Multiply by .
Multiply by .
Combine the numerators over the common denominator.
Simplify each term.
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Multiply by .