Solve 2x^4y^6=0 - ScanMath

Question

Solve the equation

Solve for x

Solve for y

x = 0

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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 θ = \frac{kπ}{2}, k \in Z

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Find the derivative with respect to x

Find the derivative with respect to y

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

Calculate

2 x^{4} y^{6} = 0

Take the derivative of both sides

\frac{d}{d x} (2 x^{4} y^{6}) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

8 x^{3} y^{6} + 12 x^{4} y^{5} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

8 x^{3} y^{6} + 12 x^{4} y^{5} \frac{d y}{d x} = 0

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

12 x^{4} y^{5} \frac{d y}{d x} = 0 - 8 x^{3} y^{6}

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

12 x^{4} y^{5} \frac{d y}{d x} = - 8 x^{3} y^{6}

Divide both sides

\frac{12 x ^{4} y ^{5} \frac{d y}{d x}}{12 x ^{4} y ^{5}} = \frac{- 8 x ^{3} y ^{6}}{12 x ^{4} y ^{5}}

Divide the numbers

\frac{d y}{d x} = \frac{- 8 x ^{3} y ^{6}}{12 x ^{4} y ^{5}}

Divide the numbers

More Steps

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

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

Calculate

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

Calculate

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

Use the commutative property to reorder the terms

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

Calculate

More Steps

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

Calculate

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

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

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{10 y}{9 x ^{2}}

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