Solve x^3 - 2*y^2 = 3

Question

Solve the equation

Solve for x

Solve for y

x = 3 3 + 2 y^{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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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{3 x ^{2}}{4 y}

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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{24 x y ^{2} - 9 x ^{4}}{16 y ^{3}}

Calculate

x^{3} - 2 \cdot y^{2} = 3

Take the derivative of both sides

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

Calculate the derivative

More Steps

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

Calculate the derivative

3 x^{2} - 4 y \frac{d y}{d x} = 0

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

- 4 y \frac{d y}{d x} = 0 - 3 x^{2}

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

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

Divide both sides

\frac{- 4 y \frac{d y}{d x}}{- 4 y} = \frac{- 3 x ^{2}}{- 4 y}

Divide the numbers

\frac{d y}{d x} = \frac{- 3 x ^{2}}{- 4 y}

Cancel out the common factor - 1

\frac{d y}{d x} = \frac{3 x ^{2}}{4 y}

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{6 x \times 4 y - 3 x ^{2} \times \frac{d}{d x} ( 4 y )}{( 4 y ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{6 x \times 4 y - 3 x ^{2} \times 4 \frac{d y}{d x}}{( 4 y ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{24 x y - 3 x ^{2} \times 4 \frac{d y}{d x}}{( 4 y ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{24 x y - 12 x ^{2} \frac{d y}{d x}}{( 4 y ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{24 x y - 12 x ^{2} \frac{d y}{d x}}{16 y ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{6 x y - 3 x ^{2} \frac{d y}{d x}}{4 y ^{2}}

Use equation \frac{d y}{d x} = \frac{3 x ^{2}}{4 y} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{6 x y - 3 x ^{2} \times \frac{3 x ^{2}}{4 y}}{4 y ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{24 x y ^{2} - 9 x ^{4}}{16 y ^{3}}

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