Calculus Examples

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 .
Set the first derivative equal to and solve for .
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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set equal to .
Set equal to and solve for .
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Set equal to .
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 by .
Multiply by .
Add and .
Multiply by .
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 by .
Multiply by .
Add and .
Multiply by .
Change the to .
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 by .
Multiply by .
Add and .
Multiply by .
Change the to .
The final answer is the combination of both solutions.
The final solution is all the values that make true.
Split into separate intervals around the values that make the first derivative or undefined.
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 .
Raise to the power of .
Multiply by .
Multiply by .
Simplify by adding and subtracting.
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Subtract from .
Add and .
The final answer is .
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 .
Raise to the power of .
Multiply by .
Multiply by .
Simplify by adding and subtracting.
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Subtract from .
Add and .
The final answer is .
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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One to any power is one.
Multiply by .
One to any power is one.
Multiply by .
Multiply by .
Simplify by subtracting numbers.
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Subtract from .
Subtract from .
The final answer is .
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 .
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 .
Since the first derivative changed signs from negative to positive around , then there is a turning point at .
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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Apply the product rule to .
Raise to the power of .
Use the Binomial Theorem.
Simplify each term.
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Raise to the power of .
Raise to the power of .
Multiply by .
Multiply by .
Raise to the power of .
Multiply by .
Apply the product rule to .
Raise to the power of .
Multiply by .
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 .
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 .
Multiply by .
Apply the product rule to .
Raise to the power of .
Multiply by .
Rewrite as .
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Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raise to the power of .
Add and .
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.
Apply the product rule to .
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 .
Multiply by .
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 .
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.
Apply the product rule to .
Raise to the power of .
Cancel the common factor of .
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Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
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 .
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.
Add and .
Subtract from .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
Find the common denominator.
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Multiply by .
Multiply and .
Multiply by .
Multiply and .
Reorder the factors of .
Multiply by .
Multiply by .
Combine the numerators over the common denominator.
Simplify each term.
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Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Simplify terms.
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Subtract from .
Subtract from .
Add and .
Add and .
Rewrite as .
Factor out of .
Factor out of .
Move the negative in front of the fraction.
Write the and coordinates in point form.
Since the first derivative changed signs from positive to negative around , then there is a turning point at .
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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Raising to any positive power yields .
Raising to any positive power yields .
Multiply by .
Raising to any positive power yields .
Multiply by .
Simplify by adding numbers.
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Add and .
Add and .
Write the and coordinates in point form.
Since the first derivative changed signs from negative to positive around , then there is a turning point at .
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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Apply the product rule to .
Raise to the power of .
Use the Binomial Theorem.
Simplify each term.
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Raise to the power of .
Raise to the power of .
Multiply by .
Raise to the power of .
Multiply by .
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 .
Multiply by .
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 .
Rewrite as .
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Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raise to the power of .
Add and .
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.
Apply the product rule to .
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 .
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 .
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.
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.
Apply the product rule to .
Raise to the power of .
Cancel the common factor of .
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Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
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 .
Move to the left of .
Combine using the product rule for radicals.
Multiply by .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Add and .
Add and .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
Find the common denominator.
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Multiply by .
Multiply and .
Multiply by .
Multiply and .
Reorder the factors of .
Multiply by .
Multiply by .
Combine the numerators over the common denominator.
Simplify each term.
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Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Simplify terms.
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Subtract from .
Subtract from .
Subtract from .
Subtract from .
Rewrite as .
Factor out of .
Factor out of .
Move the negative in front of the fraction.
Write the and coordinates in point form.
These are the turning points.
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