Solve -3x^2y - 5 = 0

Solve the equation

Solve for x

Solve for y

x = - \frac{- 15 y}{3 y} x = \frac{- 15 y}{3 y}

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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{3 5}{3 3 cos ^{2} ( θ ) sin ( 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{2 y}{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{6 y}{x ^{2}}

Calculate

- 3 x^{2} y - 5 = 0

Take the derivative of both sides

\frac{d}{d x} (- 3 x^{2} y - 5) = \frac{d}{d x} (0)

Calculate the derivative

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

Calculate the derivative

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

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

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

Add the terms

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Use the commutative property to reorder the terms

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

Any expression multiplied by 1 remains the same

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

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

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

Solution

More Steps

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

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