Find the Horizontal Tangent Line xy^2+x^2y=6

Math
Solve the equation as in terms of .
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Move to the left side of the equation by subtracting it from both sides.
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify the numerator.
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Multiply the exponents in .
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Apply the power rule and multiply exponents, .
Multiply by .
Move to the left of .
Multiply by .
Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Simplify the expression to solve for the portion of the .
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Change the to .
Factor out of .
Factor out of .
Factor out of .
Rewrite as .
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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Multiply the exponents in .
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Apply the power rule and multiply exponents, .
Multiply by .
Move to the left of .
Multiply by .
Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Change the to .
Factor out of .
Factor out of .
Factor out of .
Rewrite as .
Move the negative in front of the fraction.
The final answer is the combination of both solutions.
Set each solution of as a function of .
Because the variable in the equation has a degree greater than , use implicit differentiation to solve for the derivative .
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Differentiate both sides of the equation.
Differentiate the left side of the equation.
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By the Sum Rule, the derivative of with respect to is .
Evaluate .
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Differentiate using the Product Rule which states that is where and .
Differentiate using the chain rule, which states that is where and .
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To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
Rewrite as .
Differentiate using the Power Rule which states that is where .
Move to the left of .
Multiply by .
Evaluate .
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Differentiate using the Product Rule which states that is where and .
Rewrite as .
Differentiate using the Power Rule which states that is where .
Move to the left of .
Reorder terms.
Since is constant with respect to , the derivative of with respect to is .
Reform the equation by setting the left side equal to the right side.
Solve for .
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Move all terms not containing to the right side of the equation.
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Subtract from both sides of the equation.
Subtract from both sides of the equation.
Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Divide each term by and simplify.
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Divide each term in by .
Simplify .
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Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify .
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Simplify each term.
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Move the negative in front of the fraction.
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
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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Multiply and .
Reorder the factors of .
Combine the numerators over the common denominator.
Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Simplify the expression.
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Rewrite as .
Move the negative in front of the fraction.
Replace with .
Set the derivative equal to then solve the equation .
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Find the LCD of the terms in the equation.
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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Since contain both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.
Steps to find the LCM for are:
1. Find the LCM for the numeric part .
2. Find the LCM for the variable part .
3. Find the LCM for the compound variable part .
4. Multiply each LCM together.
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
The factor for is itself.
occurs time.
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
The factor for is itself.
occurs time.
The LCM of is the result of multiplying all factors the greatest number of times they occur in either term.
The Least Common Multiple of some numbers is the smallest number that the numbers are factors of.
Multiply each term by and simplify.
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Multiply each term in by in order to remove all the denominators from the equation.
Simplify .
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Cancel the common factor of .
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Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Multiply by by adding the exponents.
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Move .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by .
Simplify .
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Apply the distributive property.
Simplify the expression.
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Rewrite using the commutative property of multiplication.
Multiply by .
Multiply by .
Solve the equation.
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Add to both sides of the equation.
Divide each term by and simplify.
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Divide each term in by .
Simplify .
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Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify .
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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.
Solve the function at .
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Replace the variable with in the expression.
Simplify the result.
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Simplify the numerator.
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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 .
Multiply by .
Raise to the power of .
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 .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Multiply by .
Multiply and .
Multiply by .
Rewrite as .
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Factor the perfect power out of .
Factor the perfect power out of .
Rearrange the fraction .
Reorder and .
Rewrite as .
Rewrite as .
Rewrite as .
Add parentheses.
Add parentheses.
Pull terms out from under the radical.
Apply the product rule to .
One to any power is one.
Raise to the power of .
Combine and .
Combine the numerators over the common denominator.
Simplify the denominator.
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Multiply by .
Combine and .
Reduce the expression by cancelling the common factors.
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Reduce the expression by cancelling the common factors.
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Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply the numerator by the reciprocal of the denominator.
Cancel the common factor of and .
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Rewrite as .
Move the negative in front of the fraction.
Rewrite using the commutative property of multiplication.
Multiply and .
Move to the left of .
Multiply .
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Multiply by .
Multiply by .
Reorder factors in .
The final answer is .
Solve the function at .
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Replace the variable with in the expression.
Simplify the result.
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Simplify the numerator.
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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 .
Multiply by .
Raise to the power of .
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 .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Multiply by .
Multiply and .
Multiply by .
Rewrite as .
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Factor the perfect power out of .
Factor the perfect power out of .
Rearrange the fraction .
Reorder and .
Rewrite as .
Rewrite as .
Rewrite as .
Add parentheses.
Add parentheses.
Pull terms out from under the radical.
Apply the product rule to .
One to any power is one.
Raise to the power of .
Combine and .
Combine the numerators over the common denominator.
Simplify the denominator.
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Multiply by .
Combine and .
Reduce the expression by cancelling the common factors.
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Reduce the expression by cancelling the common factors.
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Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply the numerator by the reciprocal of the denominator.
Cancel the common factor of and .
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Rewrite as .
Move the negative in front of the fraction.
Rewrite using the commutative property of multiplication.
Multiply and .
Move to the left of .
Multiply .
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Multiply by .
Multiply by .
Reorder factors in .
The final answer is .
The horizontal tangent lines are
Find the Horizontal Tangent Line xy^2+x^2y=6

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