Solve x y^32x^3y =- 4

Solve the equation

Solve for x

Solve for y

x = \frac{4 2}{4 - 1 \times y} x = - \frac{4 2}{4 - 1 \times y}

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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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r = 8 - 32 csc^{4} (2 θ) r = - 8 - 32 csc^{4} (2 θ)

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{y}{x}

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Find the second derivative with respect to x

Find the second derivative with respect to y

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

Calculate

x y^{3} 2 x^{3} y = - 4

Simplify the expression

2 x^{4} y^{4} = - 4

Take the derivative of both sides

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

Calculate the derivative

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8 x^{3} y^{4} + 8 x^{4} y^{3} \frac{d y}{d x} = \frac{d}{d x} (- 4)

Calculate the derivative

8 x^{3} y^{4} + 8 x^{4} y^{3} \frac{d y}{d x} = 0

Move the expression to the right-hand side and change its sign

8 x^{4} y^{3} \frac{d y}{d x} = 0 - 8 x^{3} y^{4}

Removing 0 doesn't change the value,so remove it from the expression

8 x^{4} y^{3} \frac{d y}{d x} = - 8 x^{3} y^{4}

Divide both sides

\frac{8 x ^{4} y ^{3} \frac{d y}{d x}}{8 x ^{4} y ^{3}} = \frac{- 8 x ^{3} y ^{4}}{8 x ^{4} y ^{3}}

Divide the numbers

\frac{d y}{d x} = \frac{- 8 x ^{3} y ^{4}}{8 x ^{4} y ^{3}}

Divide the numbers

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\frac{d y}{d x} = - \frac{y}{x}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{y}{x})

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

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

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d y}{d x} \times x - y \times 1}{x ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{x \frac{d y}{d x} - y \times 1}{x ^{2}}

Any expression multiplied by 1 remains the same

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

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

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

Solution

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

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