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Calculus Examples
Step 1
Step 1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2
Evaluate .
Step 1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Multiply by .
Step 1.3
Evaluate .
Step 1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3
Multiply by .
Step 1.4
Evaluate .
Step 1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.4.2
Differentiate using the Power Rule which states that is where .
Step 1.4.3
Multiply by .
Step 2
Step 2.1
Use the quadratic formula to find the solutions.
Step 2.2
Substitute the values , , and into the quadratic formula and solve for .
Step 2.3
Simplify.
Step 2.3.1
Simplify the numerator.
Step 2.3.1.1
Raise to the power of .
Step 2.3.1.2
Multiply .
Step 2.3.1.2.1
Multiply by .
Step 2.3.1.2.2
Multiply by .
Step 2.3.1.3
Subtract from .
Step 2.3.1.4
Rewrite as .
Step 2.3.1.4.1
Factor out of .
Step 2.3.1.4.2
Rewrite as .
Step 2.3.1.5
Pull terms out from under the radical.
Step 2.3.2
Multiply by .
Step 2.3.3
Simplify .
Step 2.3.4
Move the negative in front of the fraction.
Step 2.4
Simplify the expression to solve for the portion of the .
Step 2.4.1
Simplify the numerator.
Step 2.4.1.1
Raise to the power of .
Step 2.4.1.2
Multiply .
Step 2.4.1.2.1
Multiply by .
Step 2.4.1.2.2
Multiply by .
Step 2.4.1.3
Subtract from .
Step 2.4.1.4
Rewrite as .
Step 2.4.1.4.1
Factor out of .
Step 2.4.1.4.2
Rewrite as .
Step 2.4.1.5
Pull terms out from under the radical.
Step 2.4.2
Multiply by .
Step 2.4.3
Simplify .
Step 2.4.4
Move the negative in front of the fraction.
Step 2.4.5
Change the to .
Step 2.5
Simplify the expression to solve for the portion of the .
Step 2.5.1
Simplify the numerator.
Step 2.5.1.1
Raise to the power of .
Step 2.5.1.2
Multiply .
Step 2.5.1.2.1
Multiply by .
Step 2.5.1.2.2
Multiply by .
Step 2.5.1.3
Subtract from .
Step 2.5.1.4
Rewrite as .
Step 2.5.1.4.1
Factor out of .
Step 2.5.1.4.2
Rewrite as .
Step 2.5.1.5
Pull terms out from under the radical.
Step 2.5.2
Multiply by .
Step 2.5.3
Simplify .
Step 2.5.4
Move the negative in front of the fraction.
Step 2.5.5
Change the to .
Step 2.6
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
Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
Step 4.2.1
Simplify each term.
Step 4.2.1.1
Raise to the power of .
Step 4.2.1.2
Multiply by .
Step 4.2.1.3
Multiply by .
Step 4.2.2
Simplify by adding and subtracting.
Step 4.2.2.1
Add and .
Step 4.2.2.2
Subtract from .
Step 4.2.3
The final answer is .
Step 5
Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
Step 5.2.1
Simplify each term.
Step 5.2.1.1
Raise to the power of .
Step 5.2.1.2
Multiply by .
Step 5.2.1.3
Multiply by .
Step 5.2.2
Simplify by adding and subtracting.
Step 5.2.2.1
Add and .
Step 5.2.2.2
Subtract from .
Step 5.2.3
The final answer is .
Step 6
Step 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
Step 6.2.1
Simplify each term.
Step 6.2.1.1
Raising to any positive power yields .
Step 6.2.1.2
Multiply by .
Step 6.2.1.3
Multiply by .
Step 6.2.2
Simplify by adding and subtracting.
Step 6.2.2.1
Add and .
Step 6.2.2.2
Subtract from .
Step 6.2.3
The final answer is .
Step 7
Since the first derivative changed signs from negative to positive around , then there is a turning point at .
Step 8
Step 8.1
Find to find the y-coordinate of .
Step 8.1.1
Replace the variable with in the expression.
Step 8.1.2
Simplify .
Step 8.1.2.1
Remove parentheses.
Step 8.1.2.2
Simplify each term.
Step 8.1.2.2.1
Use the power rule to distribute the exponent.
Step 8.1.2.2.1.1
Apply the product rule to .
Step 8.1.2.2.1.2
Apply the product rule to .
Step 8.1.2.2.2
Multiply by by adding the exponents.
Step 8.1.2.2.2.1
Move .
Step 8.1.2.2.2.2
Multiply by .
Step 8.1.2.2.2.2.1
Raise to the power of .
Step 8.1.2.2.2.2.2
Use the power rule to combine exponents.
Step 8.1.2.2.2.3
Add and .
Step 8.1.2.2.3
Raise to the power of .
Step 8.1.2.2.4
Multiply by .
Step 8.1.2.2.5
Raise to the power of .
Step 8.1.2.2.6
Use the Binomial Theorem.
Step 8.1.2.2.7
Simplify each term.
Step 8.1.2.2.7.1
Raise to the power of .
Step 8.1.2.2.7.2
Raise to the power of .
Step 8.1.2.2.7.3
Multiply by .
Step 8.1.2.2.7.4
Multiply by .
Step 8.1.2.2.7.5
Rewrite as .
Step 8.1.2.2.7.5.1
Use to rewrite as .
Step 8.1.2.2.7.5.2
Apply the power rule and multiply exponents, .
Step 8.1.2.2.7.5.3
Combine and .
Step 8.1.2.2.7.5.4
Cancel the common factor of .
Step 8.1.2.2.7.5.4.1
Cancel the common factor.
Step 8.1.2.2.7.5.4.2
Rewrite the expression.
Step 8.1.2.2.7.5.5
Evaluate the exponent.
Step 8.1.2.2.7.6
Multiply by .
Step 8.1.2.2.7.7
Rewrite as .
Step 8.1.2.2.7.8
Raise to the power of .
Step 8.1.2.2.7.9
Rewrite as .
Step 8.1.2.2.7.9.1
Factor out of .
Step 8.1.2.2.7.9.2
Rewrite as .
Step 8.1.2.2.7.10
Pull terms out from under the radical.
Step 8.1.2.2.8
Add and .
Step 8.1.2.2.9
Add and .
Step 8.1.2.2.10
Use the power rule to distribute the exponent.
Step 8.1.2.2.10.1
Apply the product rule to .
Step 8.1.2.2.10.2
Apply the product rule to .
Step 8.1.2.2.11
Raise to the power of .
Step 8.1.2.2.12
Multiply by .
Step 8.1.2.2.13
Raise to the power of .
Step 8.1.2.2.14
Rewrite as .
Step 8.1.2.2.15
Expand using the FOIL Method.
Step 8.1.2.2.15.1
Apply the distributive property.
Step 8.1.2.2.15.2
Apply the distributive property.
Step 8.1.2.2.15.3
Apply the distributive property.
Step 8.1.2.2.16
Simplify and combine like terms.
Step 8.1.2.2.16.1
Simplify each term.
Step 8.1.2.2.16.1.1
Multiply by .
Step 8.1.2.2.16.1.2
Move to the left of .
Step 8.1.2.2.16.1.3
Combine using the product rule for radicals.
Step 8.1.2.2.16.1.4
Multiply by .
Step 8.1.2.2.16.1.5
Rewrite as .
Step 8.1.2.2.16.1.6
Pull terms out from under the radical, assuming positive real numbers.
Step 8.1.2.2.16.2
Add and .
Step 8.1.2.2.16.3
Add and .
Step 8.1.2.2.17
Combine and .
Step 8.1.2.2.18
Move the negative in front of the fraction.
Step 8.1.2.2.19
Multiply .
Step 8.1.2.2.19.1
Multiply by .
Step 8.1.2.2.19.2
Combine and .
Step 8.1.2.3
Find the common denominator.
Step 8.1.2.3.1
Multiply by .
Step 8.1.2.3.2
Multiply by .
Step 8.1.2.3.3
Multiply by .
Step 8.1.2.3.4
Multiply by .
Step 8.1.2.3.5
Reorder the factors of .
Step 8.1.2.3.6
Multiply by .
Step 8.1.2.3.7
Multiply by .
Step 8.1.2.4
Combine the numerators over the common denominator.
Step 8.1.2.5
Simplify each term.
Step 8.1.2.5.1
Apply the distributive property.
Step 8.1.2.5.2
Multiply by .
Step 8.1.2.5.3
Multiply by .
Step 8.1.2.5.4
Apply the distributive property.
Step 8.1.2.5.5
Multiply by .
Step 8.1.2.5.6
Multiply by .
Step 8.1.2.5.7
Apply the distributive property.
Step 8.1.2.5.8
Multiply by .
Step 8.1.2.5.9
Apply the distributive property.
Step 8.1.2.5.10
Multiply by .
Step 8.1.2.5.11
Multiply by .
Step 8.1.2.6
Simplify by adding terms.
Step 8.1.2.6.1
Subtract from .
Step 8.1.2.6.2
Add and .
Step 8.1.2.6.3
Subtract from .
Step 8.1.2.6.4
Add and .
Step 8.2
Write the and coordinates in point form.
Step 9
Since the first derivative changed signs from positive to negative around , then there is a turning point at .
Step 10
Step 10.1
Find to find the y-coordinate of .
Step 10.1.1
Replace the variable with in the expression.
Step 10.1.2
Simplify .
Step 10.1.2.1
Remove parentheses.
Step 10.1.2.2
Simplify each term.
Step 10.1.2.2.1
Use the power rule to distribute the exponent.
Step 10.1.2.2.1.1
Apply the product rule to .
Step 10.1.2.2.1.2
Apply the product rule to .
Step 10.1.2.2.2
Multiply by by adding the exponents.
Step 10.1.2.2.2.1
Move .
Step 10.1.2.2.2.2
Multiply by .
Step 10.1.2.2.2.2.1
Raise to the power of .
Step 10.1.2.2.2.2.2
Use the power rule to combine exponents.
Step 10.1.2.2.2.3
Add and .
Step 10.1.2.2.3
Raise to the power of .
Step 10.1.2.2.4
Multiply by .
Step 10.1.2.2.5
Raise to the power of .
Step 10.1.2.2.6
Use the Binomial Theorem.
Step 10.1.2.2.7
Simplify each term.
Step 10.1.2.2.7.1
Raise to the power of .
Step 10.1.2.2.7.2
Raise to the power of .
Step 10.1.2.2.7.3
Multiply by .
Step 10.1.2.2.7.4
Multiply by .
Step 10.1.2.2.7.5
Multiply by .
Step 10.1.2.2.7.6
Apply the product rule to .
Step 10.1.2.2.7.7
Raise to the power of .
Step 10.1.2.2.7.8
Multiply by .
Step 10.1.2.2.7.9
Rewrite as .
Step 10.1.2.2.7.9.1
Use to rewrite as .
Step 10.1.2.2.7.9.2
Apply the power rule and multiply exponents, .
Step 10.1.2.2.7.9.3
Combine and .
Step 10.1.2.2.7.9.4
Cancel the common factor of .
Step 10.1.2.2.7.9.4.1
Cancel the common factor.
Step 10.1.2.2.7.9.4.2
Rewrite the expression.
Step 10.1.2.2.7.9.5
Evaluate the exponent.
Step 10.1.2.2.7.10
Multiply by .
Step 10.1.2.2.7.11
Apply the product rule to .
Step 10.1.2.2.7.12
Raise to the power of .
Step 10.1.2.2.7.13
Rewrite as .
Step 10.1.2.2.7.14
Raise to the power of .
Step 10.1.2.2.7.15
Rewrite as .
Step 10.1.2.2.7.15.1
Factor out of .
Step 10.1.2.2.7.15.2
Rewrite as .
Step 10.1.2.2.7.16
Pull terms out from under the radical.
Step 10.1.2.2.7.17
Multiply by .
Step 10.1.2.2.8
Add and .
Step 10.1.2.2.9
Subtract from .
Step 10.1.2.2.10
Use the power rule to distribute the exponent.
Step 10.1.2.2.10.1
Apply the product rule to .
Step 10.1.2.2.10.2
Apply the product rule to .
Step 10.1.2.2.11
Raise to the power of .
Step 10.1.2.2.12
Multiply by .
Step 10.1.2.2.13
Raise to the power of .
Step 10.1.2.2.14
Rewrite as .
Step 10.1.2.2.15
Expand using the FOIL Method.
Step 10.1.2.2.15.1
Apply the distributive property.
Step 10.1.2.2.15.2
Apply the distributive property.
Step 10.1.2.2.15.3
Apply the distributive property.
Step 10.1.2.2.16
Simplify and combine like terms.
Step 10.1.2.2.16.1
Simplify each term.
Step 10.1.2.2.16.1.1
Multiply by .
Step 10.1.2.2.16.1.2
Multiply by .
Step 10.1.2.2.16.1.3
Multiply by .
Step 10.1.2.2.16.1.4
Multiply .
Step 10.1.2.2.16.1.4.1
Multiply by .
Step 10.1.2.2.16.1.4.2
Multiply by .
Step 10.1.2.2.16.1.4.3
Raise to the power of .
Step 10.1.2.2.16.1.4.4
Raise to the power of .
Step 10.1.2.2.16.1.4.5
Use the power rule to combine exponents.
Step 10.1.2.2.16.1.4.6
Add and .
Step 10.1.2.2.16.1.5
Rewrite as .
Step 10.1.2.2.16.1.5.1
Use to rewrite as .
Step 10.1.2.2.16.1.5.2
Apply the power rule and multiply exponents, .
Step 10.1.2.2.16.1.5.3
Combine and .
Step 10.1.2.2.16.1.5.4
Cancel the common factor of .
Step 10.1.2.2.16.1.5.4.1
Cancel the common factor.
Step 10.1.2.2.16.1.5.4.2
Rewrite the expression.
Step 10.1.2.2.16.1.5.5
Evaluate the exponent.
Step 10.1.2.2.16.2
Add and .
Step 10.1.2.2.16.3
Subtract from .
Step 10.1.2.2.17
Combine and .
Step 10.1.2.2.18
Move the negative in front of the fraction.
Step 10.1.2.2.19
Multiply .
Step 10.1.2.2.19.1
Multiply by .
Step 10.1.2.2.19.2
Combine and .
Step 10.1.2.3
Find the common denominator.
Step 10.1.2.3.1
Multiply by .
Step 10.1.2.3.2
Multiply by .
Step 10.1.2.3.3
Multiply by .
Step 10.1.2.3.4
Multiply by .
Step 10.1.2.3.5
Reorder the factors of .
Step 10.1.2.3.6
Multiply by .
Step 10.1.2.3.7
Multiply by .
Step 10.1.2.4
Combine the numerators over the common denominator.
Step 10.1.2.5
Simplify each term.
Step 10.1.2.5.1
Apply the distributive property.
Step 10.1.2.5.2
Multiply by .
Step 10.1.2.5.3
Multiply by .
Step 10.1.2.5.4
Apply the distributive property.
Step 10.1.2.5.5
Multiply by .
Step 10.1.2.5.6
Multiply by .
Step 10.1.2.5.7
Apply the distributive property.
Step 10.1.2.5.8
Multiply by .
Step 10.1.2.5.9
Multiply by .
Step 10.1.2.5.10
Apply the distributive property.
Step 10.1.2.5.11
Multiply by .
Step 10.1.2.5.12
Multiply by .
Step 10.1.2.6
Simplify by adding terms.
Step 10.1.2.6.1
Subtract from .
Step 10.1.2.6.2
Add and .
Step 10.1.2.6.3
Add and .
Step 10.1.2.6.4
Subtract from .
Step 10.2
Write the and coordinates in point form.
Step 11
These are the turning points.
Step 12