Solve 9x^2=y^2 y - ScanMath

Solve the equation

Solve for x

Solve for y

x = \frac{y y}{3} x = - \frac{y 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

Not symmetry with respect to the origin

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r = 0 r = \frac{9 cos ^{2} ( θ )}{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{6 x}{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{6 y ^{3} - 72 x ^{2}}{y ^{5}}

Calculate

9 x^{2} = y^{2} y

Simplify the expression

9 x^{2} = y^{3}

Take the derivative of both sides

\frac{d}{d x} (9 x^{2}) = \frac{d}{d x} (y^{3})

Calculate the derivative

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

Calculate the derivative

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

Swap the sides of the equation

3 y^{2} \frac{d y}{d x} = 18 x

Divide both sides

\frac{3 y ^{2} \frac{d y}{d x}}{3 y ^{2}} = \frac{18 x}{3 y ^{2}}

Divide the numbers

\frac{d y}{d x} = \frac{18 x}{3 y ^{2}}

Cancel out the common factor 3

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

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

Calculate the derivative

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

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{6 y ^{2} - 12 x y \frac{d y}{d x}}{( y ^{2} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{6 y ^{2} - 12 x y \frac{d y}{d x}}{y ^{4}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{6 y - 12 x \frac{d y}{d x}}{y ^{3}}

Use equation \frac{d y}{d x} = \frac{6 x}{y ^{2}} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{6 y - 12 x \times \frac{6 x}{y ^{2}}}{y ^{3}}

Solution

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\frac{d ^{2} y}{d x ^{2}} = \frac{6 y ^{3} - 72 x ^{2}}{y ^{5}}

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