Solve x^3 x^2y xy^2 y^3=81

Solve the equation

Solve for x

Solve for y

x = \frac{3 9}{∣ y ∣} x = - \frac{3 9}{∣ 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 = \frac{6 72 sin ^{3} ( 2 θ )}{∣ sin ( 2 θ ) ∣} r = - \frac{6 72 sin ^{3} ( 2 θ )}{∣ sin ( 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^{3} x^{2} y x y^{2} y^{3} = 81

Simplify the expression

x^{6} y^{6} = 81

Take the derivative of both sides

\frac{d}{d x} (x^{6} y^{6}) = \frac{d}{d x} (81)

Calculate the derivative

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6 x^{5} y^{6} + 6 x^{6} y^{5} \frac{d y}{d x} = \frac{d}{d x} (81)

Calculate the derivative

6 x^{5} y^{6} + 6 x^{6} y^{5} \frac{d y}{d x} = 0

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

6 x^{6} y^{5} \frac{d y}{d x} = 0 - 6 x^{5} y^{6}

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

6 x^{6} y^{5} \frac{d y}{d x} = - 6 x^{5} y^{6}

Divide both sides

\frac{6 x ^{6} y ^{5} \frac{d y}{d x}}{6 x ^{6} y ^{5}} = \frac{- 6 x ^{5} y ^{6}}{6 x ^{6} y ^{5}}

Divide the numbers

\frac{d y}{d x} = \frac{- 6 x ^{5} y ^{6}}{6 x ^{6} y ^{5}}

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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