Calculus Examples

Step 1
Find the first derivative.
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Differentiate.
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By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Evaluate .
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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 .
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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.
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Since is constant with respect to , the derivative of with respect to is .
Add and .
Step 2
Set the first derivative equal to and solve for .
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Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
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Simplify the numerator.
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Raise to the power of .
Multiply .
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Multiply by .
Multiply by .
Add and .
Rewrite as .
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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 .
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Simplify the numerator.
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Raise to the power of .
Multiply .
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Multiply by .
Multiply by .
Add and .
Rewrite as .
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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 .
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Simplify the numerator.
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Raise to the power of .
Multiply .
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Multiply by .
Multiply by .
Add and .
Rewrite as .
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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.
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Raise to the power of .
Multiply by .
Multiply by .
Simplify by subtracting numbers.
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Subtract from .
Subtract from .
The final answer is .
Step 5
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Raising to any positive power yields .
Multiply by .
Multiply by .
Simplify by adding and subtracting.
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Add and .
Subtract from .
The final answer is .
Step 6
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Multiply by by adding the exponents.
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Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Add and .
Raise to the power of .
Multiply by .
Simplify by adding and subtracting.
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Add and .
Subtract from .
The final answer is .
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.
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Find to find the y-coordinate of .
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Replace the variable with in the expression.
Simplify .
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Remove parentheses.
Simplify each term.
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Use the power rule to distribute the exponent.
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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.
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Raise to the power of .
Multiply by by adding the exponents.
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Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Add and .
Raise to the power of .
Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Rewrite as .
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Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Evaluate the exponent.
Multiply .
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Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Rewrite as .
Raise to the power of .
Rewrite as .
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Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Add and .
Add and .
Cancel the common factor of and .
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Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Use the power rule to distribute the exponent.
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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 .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Rewrite as .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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Multiply by .
Multiply by .
Multiply by .
Multiply .
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Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite as .
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Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Evaluate the exponent.
Multiply by .
Add and .
Add and .
Cancel the common factor of and .
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Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply .
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Multiply by .
Combine and .
To write as a fraction with a common denominator, multiply by .
Combine fractions.
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Combine and .
Simplify the expression.
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Combine the numerators over the common denominator.
Multiply by .
Simplify the numerator.
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Apply the distributive property.
Multiply by .
Multiply by .
Add and .
Find the common denominator.
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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.
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Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Multiply by .
Simplify by adding terms.
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Add and .
Subtract from .
Add and .
Add and .
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.
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Find to find the y-coordinate of .
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Replace the variable with in the expression.
Simplify .
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Remove parentheses.
Simplify each term.
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Use the power rule to distribute the exponent.
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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.
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Raise to the power of .
Multiply by by adding the exponents.
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Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Add and .
Raise to the power of .
Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Rewrite as .
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Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Evaluate the exponent.
Multiply .
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Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Rewrite as .
Raise to the power of .
Rewrite as .
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Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Add and .
Subtract from .
Cancel the common factor of and .
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Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Use the power rule to distribute the exponent.
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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 .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Rewrite as .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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Multiply by .
Multiply by .
Multiply by .
Multiply .
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Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite as .
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Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Evaluate the exponent.
Multiply by .
Add and .
Subtract from .
Cancel the common factor of and .
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Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply .
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Multiply by .
Combine and .
To write as a fraction with a common denominator, multiply by .
Combine fractions.
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Combine and .
Simplify the expression.
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Combine the numerators over the common denominator.
Multiply by .
Simplify the numerator.
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Apply the distributive property.
Multiply by .
Multiply by .
Add and .
Find the common denominator.
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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.
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Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Multiply by .
Simplify by adding terms.
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Add and .
Subtract from .
Subtract from .
Subtract from .
Write the and coordinates in point form.
Step 11
These are the turning points.
Step 12
Enter YOUR Problem
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