Solve 2x -3y^6 = 0

Solve the equation

Solve for x

Solve for y

x = \frac{3 y ^{6}}{2}

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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{5 2 cos ( θ ) csc ( θ ) \times csc ( θ )}{5 3}

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{1}{9 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{5}{81 y ^{11}}

Calculate

2 x - 3 y^{6} = 0

Take the derivative of both sides

\frac{d}{d x} (2 x - 3 y^{6}) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

2 - 18 y^{5} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

2 - 18 y^{5} \frac{d y}{d x} = 0

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

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

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

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

Divide both sides

\frac{- 18 y ^{5} \frac{d y}{d x}}{- 18 y ^{5}} = \frac{- 2}{- 18 y ^{5}}

Divide the numbers

\frac{d y}{d x} = \frac{- 2}{- 18 y ^{5}}

Cancel out the common factor - 2

\frac{d y}{d x} = \frac{1}{9 y ^{5}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{1}{9 y ^{5}})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{1}{9 y ^{5}})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{1}{9} \times \frac{d}{d x} (\frac{1}{y ^{5}})

Rewrite the expression in exponential form

\frac{d ^{2} y}{d x ^{2}} = \frac{1}{9} \times \frac{d}{d x} (y^{- 5})

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{1}{9} (- 5 y^{- 6} \frac{d y}{d x})

Rewrite the expression

\frac{d ^{2} y}{d x ^{2}} = \frac{1}{9} (- \frac{5 \frac{d y}{d x}}{y ^{6}})

Calculate

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

Use equation \frac{d y}{d x} = \frac{1}{9 y ^{5}} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{5 \times \frac{1}{9 y ^{5}}}{9 y ^{6}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{5}{81 y ^{11}}

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