Solve x^2 - 6y^33 = 0

Solve the equation

Solve for x

Solve for y

x = 3 y 2 y x = - 3 y 2 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

Not symmetry with respect to the origin

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r = 0 r = \frac{cos ^{2} ( θ )}{18 sin ^{3} ( 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{x}{27 y ^{2}}

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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{27 y ^{3} - 2 x ^{2}}{729 y ^{5}}

Calculate

x^{2} - 6 y^{3} 3 = 0

Simplify the expression

x^{2} - 18 y^{3} = 0

Take the derivative of both sides

\frac{d}{d x} (x^{2} - 18 y^{3}) = \frac{d}{d x} (0)

Calculate the derivative

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2 x - 54 y^{2} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

2 x - 54 y^{2} \frac{d y}{d x} = 0

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

- 54 y^{2} \frac{d y}{d x} = 0 - 2 x

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

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

Divide both sides

\frac{- 54 y ^{2} \frac{d y}{d x}}{- 54 y ^{2}} = \frac{- 2 x}{- 54 y ^{2}}

Divide the numbers

\frac{d y}{d x} = \frac{- 2 x}{- 54 y ^{2}}

Cancel out the common factor - 2

\frac{d y}{d x} = \frac{x}{27 y ^{2}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{x}{27 y ^{2}})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{x}{27 y ^{2}})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( x ) \times 27 y ^{2} - x \times \frac{d}{d x} ( 27 y ^{2} )}{( 27 y ^{2} ) ^{2}}

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

\frac{d ^{2} y}{d x ^{2}} = \frac{1 \times 27 y ^{2} - x \times \frac{d}{d x} ( 27 y ^{2} )}{( 27 y ^{2} ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{1 \times 27 y ^{2} - x \times 54 y \frac{d y}{d x}}{( 27 y ^{2} ) ^{2}}

Any expression multiplied by 1 remains the same

\frac{d ^{2} y}{d x ^{2}} = \frac{27 y ^{2} - x \times 54 y \frac{d y}{d x}}{( 27 y ^{2} ) ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = \frac{27 y ^{2} - 54 x y \frac{d y}{d x}}{( 27 y ^{2} ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{27 y ^{2} - 54 x y \frac{d y}{d x}}{729 y ^{4}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{y - 2 x \frac{d y}{d x}}{27 y ^{3}}

Use equation \frac{d y}{d x} = \frac{x}{27 y ^{2}} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{y - 2 x \times \frac{x}{27 y ^{2}}}{27 y ^{3}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{27 y ^{3} - 2 x ^{2}}{729 y ^{5}}

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