Solve \frac{8x}{4y}=\frac{1}{3}

Solve the equation

Solve for x

Solve for y

x = \frac{y}{6}

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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 = arctan (6) + kπ, 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{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}} = 0

Calculate

\frac{8 x}{4 y} = \frac{1}{3}

Simplify the expression

\frac{2 x}{y} = \frac{1}{3}

Take the derivative of both sides

\frac{d}{d x} (\frac{2 x}{y}) = \frac{d}{d x} (\frac{1}{3})

Calculate the derivative

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

Calculate the derivative

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

Simplify

2 y - 2 x \frac{d y}{d x} = 0

Move the constant to the right side

- 2 x \frac{d y}{d x} = 0 - 2 y

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

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

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

Use the commutative property to reorder the terms

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

Any expression multiplied by 1 remains the same

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

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

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

Solution

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\frac{d ^{2} y}{d x ^{2}} = 0

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