Solve 2y^2 x y 1=x^22y^2 xy

Question

Solve the equation

Solve for x

Solve for y

x = 0 x = 1 x = - 1

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Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Symmetry with respect to the origin

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Rewrite the equation

r = 0 r = ∣ sec (θ) ∣ r = - ∣ sec (θ) ∣

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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{3 y x ^{2} - y}{3 x - 3 x ^{3}}

Calculate

2 y^{2} x y 1 = x^{2} 2 y^{2} x y

Simplify the expression

2 y^{3} x = 2 x^{3} y^{3}

Take the derivative of both sides

\frac{d}{d x} (2 y^{3} x) = \frac{d}{d x} (2 x^{3} y^{3})

Calculate the derivative

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2 y^{3} + 6 x y^{2} \frac{d y}{d x} = \frac{d}{d x} (2 x^{3} y^{3})

Calculate the derivative

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2 y^{3} + 6 x y^{2} \frac{d y}{d x} = 6 x^{2} y^{3} + 6 x^{3} y^{2} \frac{d y}{d x}

Move the expression to the left side

2 y^{3} + 6 x y^{2} \frac{d y}{d x} - 6 x^{3} y^{2} \frac{d y}{d x} = 6 x^{2} y^{3}

Move the expression to the right side

6 x y^{2} \frac{d y}{d x} - 6 x^{3} y^{2} \frac{d y}{d x} = 6 x^{2} y^{3} - 2 y^{3}

Collect like terms by calculating the sum or difference of their coefficients

(6 x y^{2} - 6 x^{3} y^{2}) \frac{d y}{d x} = 6 x^{2} y^{3} - 2 y^{3}

Divide both sides

\frac{( 6 x y ^{2} - 6 x ^{3} y ^{2} ) \frac{d y}{d x}}{6 x y ^{2} - 6 x ^{3} y ^{2}} = \frac{6 x ^{2} y ^{3} - 2 y ^{3}}{6 x y ^{2} - 6 x ^{3} y ^{2}}

Divide the numbers

\frac{d y}{d x} = \frac{6 x ^{2} y ^{3} - 2 y ^{3}}{6 x y ^{2} - 6 x ^{3} y ^{2}}

Solution

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\frac{d y}{d x} = \frac{3 y x ^{2} - y}{3 x - 3 x ^{3}}

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Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = \frac{18 x ^{4} y - 6 x ^{2} y + 4 y}{9 x ^{2} - 18 x ^{4} + 9 x ^{6}}

Calculate

2 y^{2} x y 1 = x^{2} 2 y^{2} x y

Simplify the expression

2 y^{3} x = 2 x^{3} y^{3}

Take the derivative of both sides

\frac{d}{d x} (2 y^{3} x) = \frac{d}{d x} (2 x^{3} y^{3})

Calculate the derivative

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2 y^{3} + 6 x y^{2} \frac{d y}{d x} = \frac{d}{d x} (2 x^{3} y^{3})

Calculate the derivative

More Steps

2 y^{3} + 6 x y^{2} \frac{d y}{d x} = 6 x^{2} y^{3} + 6 x^{3} y^{2} \frac{d y}{d x}

Move the expression to the left side

2 y^{3} + 6 x y^{2} \frac{d y}{d x} - 6 x^{3} y^{2} \frac{d y}{d x} = 6 x^{2} y^{3}

Move the expression to the right side

6 x y^{2} \frac{d y}{d x} - 6 x^{3} y^{2} \frac{d y}{d x} = 6 x^{2} y^{3} - 2 y^{3}

Collect like terms by calculating the sum or difference of their coefficients

(6 x y^{2} - 6 x^{3} y^{2}) \frac{d y}{d x} = 6 x^{2} y^{3} - 2 y^{3}

Divide both sides

\frac{( 6 x y ^{2} - 6 x ^{3} y ^{2} ) \frac{d y}{d x}}{6 x y ^{2} - 6 x ^{3} y ^{2}} = \frac{6 x ^{2} y ^{3} - 2 y ^{3}}{6 x y ^{2} - 6 x ^{3} y ^{2}}

Divide the numbers

\frac{d y}{d x} = \frac{6 x ^{2} y ^{3} - 2 y ^{3}}{6 x y ^{2} - 6 x ^{3} y ^{2}}

Divide the numbers

More Steps

\frac{d y}{d x} = \frac{3 y x ^{2} - y}{3 x - 3 x ^{3}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{3 y x ^{2} - y}{3 x - 3 x ^{3}})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{3 y x ^{2} - y}{3 x - 3 x ^{3}})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 3 y x ^{2} - y ) \times ( 3 x - 3 x ^{3} ) - ( 3 y x ^{2} - y ) \times \frac{d}{d x} ( 3 x - 3 x ^{3} )}{( 3 x - 3 x ^{3} ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( 6 x y + 3 x ^{2} \frac{d y}{d x} - \frac{d y}{d x} ) ( 3 x - 3 x ^{3} ) - ( 3 y x ^{2} - y ) \times \frac{d}{d x} ( 3 x - 3 x ^{3} )}{( 3 x - 3 x ^{3} ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{( 6 x y + 3 x ^{2} \frac{d y}{d x} - \frac{d y}{d x} ) ( 3 x - 3 x ^{3} ) - ( 3 y x ^{2} - y ) ( 3 - 9 x ^{2} )}{( 3 x - 3 x ^{3} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{18 x ^{2} y - 18 x ^{4} y + 12 x ^{3} \frac{d y}{d x} - 9 x ^{5} \frac{d y}{d x} - 3 x \frac{d y}{d x} - ( 3 y x ^{2} - y ) ( 3 - 9 x ^{2} )}{( 3 x - 3 x ^{3} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{18 x ^{2} y - 18 x ^{4} y + 12 x ^{3} \frac{d y}{d x} - 9 x ^{5} \frac{d y}{d x} - 3 x \frac{d y}{d x} - ( 18 y x ^{2} - 3 y - 27 y x ^{4} )}{( 3 x - 3 x ^{3} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{9 x ^{4} y + 12 x ^{3} \frac{d y}{d x} - 9 x ^{5} \frac{d y}{d x} - 3 x \frac{d y}{d x} + 3 y}{( 3 x - 3 x ^{3} ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{3 x ^{4} y + 4 x ^{3} \frac{d y}{d x} - 3 x ^{5} \frac{d y}{d x} - x \frac{d y}{d x} + y}{3 ( x - x ^{3} ) ^{2}}

Use equation \frac{d y}{d x} = \frac{3 y x ^{2} - y}{3 x - 3 x ^{3}} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{3 x ^{4} y + 4 x ^{3} \times \frac{3 y x ^{2} - y}{3 x - 3 x ^{3}} - 3 x ^{5} \times \frac{3 y x ^{2} - y}{3 x - 3 x ^{3}} - x \times \frac{3 y x ^{2} - y}{3 x - 3 x ^{3}} + y}{3 ( x - x ^{3} ) ^{2}}

Solution

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Calculate

\frac{3 x ^{4} y + 4 x ^{3} \times \frac{3 y x ^{2} - y}{3 x - 3 x ^{3}} - 3 x ^{5} \times \frac{3 y x ^{2} - y}{3 x - 3 x ^{3}} - x \times \frac{3 y x ^{2} - y}{3 x - 3 x ^{3}} + y}{3 ( x - x ^{3} ) ^{2}}

Multiply the terms

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\frac{3 x ^{4} y + \frac{4 x ^{2} ( 3 y x ^{2} - y )}{3 - 3 x ^{2}} - 3 x ^{5} \times \frac{3 y x ^{2} - y}{3 x - 3 x ^{3}} - x \times \frac{3 y x ^{2} - y}{3 x - 3 x ^{3}} + y}{3 ( x - x ^{3} ) ^{2}}

Multiply the terms

\frac{3 x ^{4} y + \frac{4 x ^{2} ( 3 y x ^{2} - y )}{3 - 3 x ^{2}} - \frac{x ^{4} ( 3 y x ^{2} - y )}{1 - x ^{2}} - x \times \frac{3 y x ^{2} - y}{3 x - 3 x ^{3}} + y}{3 ( x - x ^{3} ) ^{2}}

Multiply the terms

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\frac{3 x ^{4} y + \frac{4 x ^{2} ( 3 y x ^{2} - y )}{3 - 3 x ^{2}} - \frac{x ^{4} ( 3 y x ^{2} - y )}{1 - x ^{2}} - \frac{3 y x ^{2} - y}{3 - 3 x ^{2}} + y}{3 ( x - x ^{3} ) ^{2}}

Calculate the sum or difference

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Evaluate

3 x^{4} y + \frac{4 x ^{2} ( 3 y x ^{2} - y )}{3 - 3 x ^{2}} - \frac{x ^{4} ( 3 y x ^{2} - y )}{1 - x ^{2}} - \frac{3 y x ^{2} - y}{3 - 3 x ^{2}} + y

Rewrite the fractions

3 x^{4} y + \frac{4 x ^{2} ( 3 y x ^{2} - y )}{3 - 3 x ^{2}} + \frac{x ^{4} ( 3 y x ^{2} - y )}{- 1 + x ^{2}} - \frac{3 y x ^{2} - y}{3 - 3 x ^{2}} + y

Rewrite the fractions

3 x^{4} y + \frac{4 x ^{2} ( 3 y x ^{2} - y )}{3 - 3 x ^{2}} + \frac{x ^{4} ( 3 y x ^{2} - y )}{- 1 + x ^{2}} + \frac{3 y x ^{2} - y}{- 3 + 3 x ^{2}} + y

Factor the expression

3 x^{4} y + \frac{4 x ^{2} ( 3 y x ^{2} - y )}{3 - 3 x ^{2}} + \frac{x ^{4} ( 3 y x ^{2} - y )}{- 1 + x ^{2}} + \frac{3 y x ^{2} - y}{3 ( x ^{2} - 1 )} + y

Reduce fractions to a common denominator

\frac{3 x ^{4} y ( 3 - 3 x ^{2} )}{3 - 3 x ^{2}} + \frac{4 x ^{2} ( 3 y x ^{2} - y )}{3 - 3 x ^{2}} + \frac{x ^{4} ( 3 y x ^{2} - y ) ( - 3 )}{( - 1 + x ^{2} ) ( - 3 )} + \frac{( 3 y x ^{2} - y ) ( - 1 )}{3 ( x ^{2} - 1 ) ( - 1 )} + \frac{y ( - 3 ) ( - 1 ) ( - 1 ) ( x ^{2} - 1 )}{- 3 ( - 1 ) ( - 1 ) ( x ^{2} - 1 )}

Use the commutative property to reorder the terms

\frac{3 x ^{4} y ( 3 - 3 x ^{2} )}{3 - 3 x ^{2}} + \frac{4 x ^{2} ( 3 y x ^{2} - y )}{3 - 3 x ^{2}} + \frac{x ^{4} ( 3 y x ^{2} - y ) ( - 3 )}{- 3 ( - 1 + x ^{2} )} + \frac{( 3 y x ^{2} - y ) ( - 1 )}{3 ( x ^{2} - 1 ) ( - 1 )} + \frac{y ( - 3 ) ( - 1 ) ( - 1 ) ( x ^{2} - 1 )}{- 3 ( - 1 ) ( - 1 ) ( x ^{2} - 1 )}

Rewrite the expression

\frac{3 x ^{4} y ( 3 - 3 x ^{2} )}{3 - 3 x ^{2}} + \frac{4 x ^{2} ( 3 y x ^{2} - y )}{3 - 3 x ^{2}} + \frac{x ^{4} ( 3 y x ^{2} - y ) ( - 3 )}{- 3 ( - 1 + x ^{2} )} + \frac{( 3 y x ^{2} - y ) ( - 1 )}{- 3 ( x ^{2} - 1 )} + \frac{y ( - 3 ) ( - 1 ) ( - 1 ) ( x ^{2} - 1 )}{- 3 ( - 1 ) ( - 1 ) ( x ^{2} - 1 )}

Rewrite the expression

\frac{3 x ^{4} y ( 3 - 3 x ^{2} )}{3 - 3 x ^{2}} + \frac{4 x ^{2} ( 3 y x ^{2} - y )}{3 - 3 x ^{2}} + \frac{x ^{4} ( 3 y x ^{2} - y ) ( - 3 )}{- 3 ( - 1 + x ^{2} )} + \frac{( 3 y x ^{2} - y ) ( - 1 )}{- 3 ( x ^{2} - 1 )} + \frac{y ( - 3 ) ( - 1 ) ( - 1 ) ( x ^{2} - 1 )}{- 3 ( x ^{2} - 1 )}

Rewrite the expression

\frac{3 x ^{4} y ( 3 - 3 x ^{2} )}{3 - 3 x ^{2}} + \frac{4 x ^{2} ( 3 y x ^{2} - y )}{3 - 3 x ^{2}} + \frac{x ^{4} ( 3 y x ^{2} - y ) ( - 3 )}{3 - 3 x ^{2}} + \frac{( 3 y x ^{2} - y ) ( - 1 )}{3 - 3 x ^{2}} + \frac{y ( - 3 ) ( - 1 ) ( - 1 ) ( x ^{2} - 1 )}{3 - 3 x ^{2}}

Write all numerators above the common denominator

\frac{3 x ^{4} y ( 3 - 3 x ^{2} ) + 4 x ^{2} ( 3 y x ^{2} - y ) + x ^{4} ( 3 y x ^{2} - y ) ( - 3 ) + ( 3 y x ^{2} - y ) ( - 1 ) + y ( - 3 ) ( - 1 ) ( - 1 ) ( x ^{2} - 1 )}{3 - 3 x ^{2}}

Multiply the terms

\frac{9 x ^{4} y - 9 x ^{6} y + 4 x ^{2} ( 3 y x ^{2} - y ) + x ^{4} ( 3 y x ^{2} - y ) ( - 3 ) + ( 3 y x ^{2} - y ) ( - 1 ) + y ( - 3 ) ( - 1 ) ( - 1 ) ( x ^{2} - 1 )}{3 - 3 x ^{2}}

Multiply the terms

\frac{9 x ^{4} y - 9 x ^{6} y + 12 y x ^{4} - 4 y x ^{2} + x ^{4} ( 3 y x ^{2} - y ) ( - 3 ) + ( 3 y x ^{2} - y ) ( - 1 ) + y ( - 3 ) ( - 1 ) ( - 1 ) ( x ^{2} - 1 )}{3 - 3 x ^{2}}

Multiply the terms

\frac{9 x ^{4} y - 9 x ^{6} y + 12 y x ^{4} - 4 y x ^{2} - 9 x ^{6} y + 3 x ^{4} y + ( 3 y x ^{2} - y ) ( - 1 ) + y ( - 3 ) ( - 1 ) ( - 1 ) ( x ^{2} - 1 )}{3 - 3 x ^{2}}

Multiply the terms

\frac{9 x ^{4} y - 9 x ^{6} y + 12 y x ^{4} - 4 y x ^{2} - 9 x ^{6} y + 3 x ^{4} y - 3 y x ^{2} + y + y ( - 3 ) ( - 1 ) ( - 1 ) ( x ^{2} - 1 )}{3 - 3 x ^{2}}

Multiply the terms

\frac{9 x ^{4} y - 9 x ^{6} y + 12 y x ^{4} - 4 y x ^{2} - 9 x ^{6} y + 3 x ^{4} y - 3 y x ^{2} + y - 3 y x ^{2} + 3 y}{3 - 3 x ^{2}}

Calculate the sum or difference

\frac{24 x ^{4} y - 18 x ^{6} y - 10 y x ^{2} + 4 y}{3 - 3 x ^{2}}

Factor the expression

\frac{( x ^{2} - 1 ) ( - 18 x ^{4} y + 6 x ^{2} y - 4 y )}{3 - 3 x ^{2}}

Factor the expression

\frac{( x ^{2} - 1 ) ( - 18 x ^{4} y + 6 x ^{2} y - 4 y )}{- 3 ( x ^{2} - 1 )}

Reduce the fraction

\frac{- 18 x ^{4} y + 6 x ^{2} y - 4 y}{- 3}

Calculate

\frac{18 x ^{4} y - 6 x ^{2} y + 4 y}{3}

\frac{\frac{18 x ^{4} y - 6 x ^{2} y + 4 y}{3}}{3 ( x - x ^{3} ) ^{2}}

Multiply by the reciprocal

\frac{18 x ^{4} y - 6 x ^{2} y + 4 y}{3} \times \frac{1}{3 ( x - x ^{3} ) ^{2}}

Multiply the terms

\frac{18 x ^{4} y - 6 x ^{2} y + 4 y}{3 \times 3 ( x - x ^{3} ) ^{2}}

Multiply the terms

\frac{18 x ^{4} y - 6 x ^{2} y + 4 y}{9 ( x - x ^{3} ) ^{2}}

Expand the expression

More Steps

\frac{18 x ^{4} y - 6 x ^{2} y + 4 y}{9 x ^{2} - 18 x ^{4} + 9 x ^{6}}

\frac{d ^{2} y}{d x ^{2}} = \frac{18 x ^{4} y - 6 x ^{2} y + 4 y}{9 x ^{2} - 18 x ^{4} + 9 x ^{6}}

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