Solve x^2 3xy 10=8x^6y

Question

Solve the equation

Solve for x

Solve for y

x = 0 x = \frac{3 30}{2}

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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

Not symmetry with respect to the origin

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Rewrite the equation

r = 0 r = \frac{3 15}{3 4 \times cos ( 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{24 x ^{3} y - 45 y}{15 x - 4 x ^{4}}

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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{- 2160 x ^{3} y + 672 x ^{6} y + 2700 y}{225 x ^{2} - 120 x ^{5} + 16 x ^{8}}

Calculate

x^{2} 3 x y 10 = 8 x^{6} y

Simplify the expression

30 x^{3} y = 8 x^{6} y

Take the derivative of both sides

\frac{d}{d x} (30 x^{3} y) = \frac{d}{d x} (8 x^{6} y)

Calculate the derivative

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90 x^{2} y + 30 x^{3} \frac{d y}{d x} = \frac{d}{d x} (8 x^{6} y)

Calculate the derivative

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90 x^{2} y + 30 x^{3} \frac{d y}{d x} = 48 x^{5} y + 8 x^{6} \frac{d y}{d x}

Move the expression to the left side

90 x^{2} y + 30 x^{3} \frac{d y}{d x} - 8 x^{6} \frac{d y}{d x} = 48 x^{5} y

Move the expression to the right side

30 x^{3} \frac{d y}{d x} - 8 x^{6} \frac{d y}{d x} = 48 x^{5} y - 90 x^{2} y

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

(30 x^{3} - 8 x^{6}) \frac{d y}{d x} = 48 x^{5} y - 90 x^{2} y

Divide both sides

\frac{( 30 x ^{3} - 8 x ^{6} ) \frac{d y}{d x}}{30 x ^{3} - 8 x ^{6}} = \frac{48 x ^{5} y - 90 x ^{2} y}{30 x ^{3} - 8 x ^{6}}

Divide the numbers

\frac{d y}{d x} = \frac{48 x ^{5} y - 90 x ^{2} y}{30 x ^{3} - 8 x ^{6}}

Divide the numbers

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\frac{d y}{d x} = \frac{24 x ^{3} y - 45 y}{15 x - 4 x ^{4}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{24 x ^{3} y - 45 y}{15 x - 4 x ^{4}})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{24 x ^{3} y - 45 y}{15 x - 4 x ^{4}})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 24 x ^{3} y - 45 y ) \times ( 15 x - 4 x ^{4} ) - ( 24 x ^{3} y - 45 y ) \times \frac{d}{d x} ( 15 x - 4 x ^{4} )}{( 15 x - 4 x ^{4} ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( 72 x ^{2} y + 24 x ^{3} \frac{d y}{d x} - 45 \frac{d y}{d x} ) ( 15 x - 4 x ^{4} ) - ( 24 x ^{3} y - 45 y ) \times \frac{d}{d x} ( 15 x - 4 x ^{4} )}{( 15 x - 4 x ^{4} ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( 72 x ^{2} y + 24 x ^{3} \frac{d y}{d x} - 45 \frac{d y}{d x} ) ( 15 x - 4 x ^{4} ) - ( 24 x ^{3} y - 45 y ) ( 15 - 16 x ^{3} )}{( 15 x - 4 x ^{4} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{1080 x ^{3} y - 288 x ^{6} y + 540 x ^{4} \frac{d y}{d x} - 96 x ^{7} \frac{d y}{d x} - 675 x \frac{d y}{d x} - ( 24 x ^{3} y - 45 y ) ( 15 - 16 x ^{3} )}{( 15 x - 4 x ^{4} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{1080 x ^{3} y - 288 x ^{6} y + 540 x ^{4} \frac{d y}{d x} - 96 x ^{7} \frac{d y}{d x} - 675 x \frac{d y}{d x} - ( 1080 x ^{3} y - 675 y - 384 x ^{6} y )}{( 15 x - 4 x ^{4} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{96 x ^{6} y + 540 x ^{4} \frac{d y}{d x} - 96 x ^{7} \frac{d y}{d x} - 675 x \frac{d y}{d x} + 675 y}{( 15 x - 4 x ^{4} ) ^{2}}

Use equation \frac{d y}{d x} = \frac{24 x ^{3} y - 45 y}{15 x - 4 x ^{4}} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{96 x ^{6} y + 540 x ^{4} \times \frac{24 x ^{3} y - 45 y}{15 x - 4 x ^{4}} - 96 x ^{7} \times \frac{24 x ^{3} y - 45 y}{15 x - 4 x ^{4}} - 675 x \times \frac{24 x ^{3} y - 45 y}{15 x - 4 x ^{4}} + 675 y}{( 15 x - 4 x ^{4} ) ^{2}}

Solution

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Calculate

\frac{96 x ^{6} y + 540 x ^{4} \times \frac{24 x ^{3} y - 45 y}{15 x - 4 x ^{4}} - 96 x ^{7} \times \frac{24 x ^{3} y - 45 y}{15 x - 4 x ^{4}} - 675 x \times \frac{24 x ^{3} y - 45 y}{15 x - 4 x ^{4}} + 675 y}{( 15 x - 4 x ^{4} ) ^{2}}

Multiply the terms

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\frac{96 x ^{6} y + \frac{540 x ^{3} ( 24 x ^{3} y - 45 y )}{15 - 4 x ^{3}} - 96 x ^{7} \times \frac{24 x ^{3} y - 45 y}{15 x - 4 x ^{4}} - 675 x \times \frac{24 x ^{3} y - 45 y}{15 x - 4 x ^{4}} + 675 y}{( 15 x - 4 x ^{4} ) ^{2}}

Multiply the terms

\frac{96 x ^{6} y + \frac{540 x ^{3} ( 24 x ^{3} y - 45 y )}{15 - 4 x ^{3}} - \frac{96 x ^{6} ( 24 x ^{3} y - 45 y )}{15 - 4 x ^{3}} - 675 x \times \frac{24 x ^{3} y - 45 y}{15 x - 4 x ^{4}} + 675 y}{( 15 x - 4 x ^{4} ) ^{2}}

Multiply the terms

\frac{96 x ^{6} y + \frac{540 x ^{3} ( 24 x ^{3} y - 45 y )}{15 - 4 x ^{3}} - \frac{96 x ^{6} ( 24 x ^{3} y - 45 y )}{15 - 4 x ^{3}} - \frac{675 ( 24 x ^{3} y - 45 y )}{15 - 4 x ^{3}} + 675 y}{( 15 x - 4 x ^{4} ) ^{2}}

Calculate the sum or difference

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\frac{- 2160 x ^{3} y + 672 x ^{6} y + 2700 y}{( 15 x - 4 x ^{4} ) ^{2}}

Expand the expression

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\frac{- 2160 x ^{3} y + 672 x ^{6} y + 2700 y}{225 x ^{2} - 120 x ^{5} + 16 x ^{8}}

\frac{d ^{2} y}{d x ^{2}} = \frac{- 2160 x ^{3} y + 672 x ^{6} y + 2700 y}{225 x ^{2} - 120 x ^{5} + 16 x ^{8}}

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