Solve x^22y^22xy y=2

Question

Solve the equation

Solve for x

Solve for y

x = \frac{3 4 y ^{2}}{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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r = \frac{7 csc ^{4} ( θ ) sec ^{3} ( θ )}{7 2}

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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 y}{4 x}

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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{21 y}{16 x ^{2}}

Calculate

x^{2} 2 y^{2} 2 x yy = 2

Simplify the expression

4 x^{3} y^{4} = 2

Take the derivative of both sides

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

Calculate the derivative

More Steps

12 x^{2} y^{4} + 16 x^{3} y^{3} \frac{d y}{d x} = \frac{d}{d x} (2)

Calculate the derivative

12 x^{2} y^{4} + 16 x^{3} y^{3} \frac{d y}{d x} = 0

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

16 x^{3} y^{3} \frac{d y}{d x} = 0 - 12 x^{2} y^{4}

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

16 x^{3} y^{3} \frac{d y}{d x} = - 12 x^{2} y^{4}

Divide both sides

\frac{16 x ^{3} y ^{3} \frac{d y}{d x}}{16 x ^{3} y ^{3}} = \frac{- 12 x ^{2} y ^{4}}{16 x ^{3} y ^{3}}

Divide the numbers

\frac{d y}{d x} = \frac{- 12 x ^{2} y ^{4}}{16 x ^{3} y ^{3}}

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

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

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{12 \frac{d y}{d x} \times x - 3 y \times 4}{( 4 x ) ^{2}}

Calculate

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

Use the commutative property to reorder the terms

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

Calculate

More Steps

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

Calculate

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

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

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{21 y}{16 x ^{2}}

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