Solve x^2 - y^6 = 0

Solve the equation

Solve for x

Solve for y

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

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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 = 0 r = 4 \frac{cos ^{2} ( θ )}{sin ^{6} ( θ )} r = - 4 \frac{cos ^{2} ( θ )}{sin ^{6} ( 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}{3 y ^{5}}

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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{3 y ^{6} - 5 x ^{2}}{9 y ^{11}}

Calculate

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

Take the derivative of both sides

\frac{d}{d x} (x^{2} - y^{6}) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

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

Calculate the derivative

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

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

- 6 y^{5} \frac{d y}{d x} = 0 - 2 x

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

- 6 y^{5} \frac{d y}{d x} = - 2 x

Divide both sides

\frac{- 6 y ^{5} \frac{d y}{d x}}{- 6 y ^{5}} = \frac{- 2 x}{- 6 y ^{5}}

Divide the numbers

\frac{d y}{d x} = \frac{- 2 x}{- 6 y ^{5}}

Cancel out the common factor - 2

\frac{d y}{d x} = \frac{x}{3 y ^{5}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{x}{3 y ^{5}})

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{1 \times 3 y ^{5} - x \times 15 y ^{4} \frac{d y}{d x}}{( 3 y ^{5} ) ^{2}}

Any expression multiplied by 1 remains the same

\frac{d ^{2} y}{d x ^{2}} = \frac{3 y ^{5} - x \times 15 y ^{4} \frac{d y}{d x}}{( 3 y ^{5} ) ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = \frac{3 y ^{5} - 15 x y ^{4} \frac{d y}{d x}}{( 3 y ^{5} ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{3 y ^{5} - 15 x y ^{4} \frac{d y}{d x}}{9 y ^{10}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{y - 5 x \frac{d y}{d x}}{3 y ^{6}}

Use equation \frac{d y}{d x} = \frac{x}{3 y ^{5}} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{y - 5 x \times \frac{x}{3 y ^{5}}}{3 y ^{6}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{3 y ^{6} - 5 x ^{2}}{9 y ^{11}}

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