Solve -2x^3y = -10

Solve the equation

$Solve for x$

$Solve for y$

x = \frac{3 5 y ^{2}}{y}

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Testing for symmetry

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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Rewrite the equation

Rewrite in polar form

r = 4 5 sec^{3} (θ) csc (θ) r = - 4 5 sec^{3} (θ) csc (θ)

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Find the first derivative

$Find the derivative with respect to x$

$Find the derivative with respect to y$

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

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Find the second derivative

$Find the second derivative with respect to x$

$Find the second derivative with respect to y$

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

Calculate

- 2 x^{3} y = - 10

Take the derivative of both sides

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

Calculate the derivative

More Steps

- 6 x^{2} y - 2 x^{3} \frac{d y}{d x} = \frac{d}{d x} (- 10)

Calculate the derivative

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

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

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

Add the terms

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

More Steps

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

\frac{d ^{2} y}{d x ^{2}} = - \frac{3 \frac{d y}{d x} \times x - 3 y \times 1}{x ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{3 x \frac{d y}{d x} - 3 y \times 1}{x ^{2}}

Any expression multiplied by 1 remains the same

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

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

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

Solution

More Steps

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

Show Solution

2x^3y = 10

Solve the equation

Solve for x

Solve for y

Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Rewrite the equation

Rewrite in polar form

Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

$Solve for x$

$Solve for y$

$Find the derivative with respect to x$

$Find the derivative with respect to y$

$Find the second derivative with respect to x$

$Find the second derivative with respect to y$