Solve |x|=|y| - ScanMath

Question

Solve the equation

Solve for x

Solve for y

x = y x = - 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

Symmetry with respect to the origin

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{x ∣ y ∣}{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

∣ x ∣ = ∣ y ∣

Take the derivative of both sides

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

Calculate the derivative

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

Calculate the derivative

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

Swap the sides of the equation

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

Multiply both sides of the equation by ∣ y ∣

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

Multiply the terms

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

Multiply by the reciprocal

y \frac{d y}{d x} \times \frac{1}{y} = \frac{x ∣ y ∣}{∣ x ∣} \times \frac{1}{y}

Multiply

\frac{d y}{d x} = \frac{x ∣ y ∣}{∣ x ∣} \times \frac{1}{y}

Multiply

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

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

Calculate the derivative

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

Calculate

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

Calculate

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

Calculate

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

Calculate

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

Solution

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

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