Calculus Examples

Find Where dy/dx is Equal to Zero
Differentiate both sides of the equation.
The derivative of with respect to is .
Differentiate the right side of the equation.
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Differentiate using the chain rule, which states that is where and .
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To apply the Chain Rule, set as .
The derivative of with respect to is .
Replace all occurrences of with .
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 .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Simplify the expression.
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Multiply by .
Reorder terms.
Reform the equation by setting the left side equal to the right side.
Simplify.
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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Multiply and .
Reorder factors in .
Replace with .
Set then solve for in terms of .
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Multiply each term by and simplify.
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Multiply each term in by .
Simplify the left side of the equation by cancelling the common factors.
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Cancel the common factor of .
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Write as a fraction with denominator .
Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Simplify.
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Multiply and .
Divide by .
Simplify terms.
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Apply the distributive property.
Reorder.
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Rewrite using the commutative property of multiplication.
Move to the left of .
Simplify each term.
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Multiply by by adding the exponents.
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Move .
Multiply by .
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 by adding the exponents.
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Move .
Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Add and .
Multiply by .
Multiply by by adding the exponents.
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Move .
Multiply by .
Multiply .
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Multiply by .
Multiply by .
Subtract from .
Multiply by .
Factor the left side of the equation.
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Factor out of .
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Factor out of .
Factor out of .
Raise to the power of .
Factor out of .
Factor out of .
Factor out of .
Factor.
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Factor by grouping.
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For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Factor out of .
Rewrite as plus
Apply the distributive property.
Factor out the greatest common factor from each group.
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Group the first two terms and the last two terms.
Factor out the greatest common factor (GCF) from each group.
Factor the polynomial by factoring out the greatest common factor, .
Remove unnecessary parentheses.
Set equal to and solve for .
Set equal to and solve for .
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Set the factor equal to .
Add to both sides of the equation.
Set equal to and solve for .
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Set the factor equal to .
Add to both sides of the equation.
Divide each term by and simplify.
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Divide each term in by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
The solution is the result of and .
Exclude the solutions that do not make true.
Simplify .
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Simplify each term.
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Apply the product rule to .
One to any power is one.
Raise to the power of .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Move the negative in front of the fraction.
is approximately which is negative so negate and remove the absolute value
Find the points where .
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