Solve x^2+(y\sqrt 2x)2=1

Question

Solve the equation

y = \frac{2 x - x ^{2} 2 x}{4 x}

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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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Find the derivative with respect to x

Find the derivative with respect to y

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

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\frac{d ^{2} y}{d x ^{2}} = \frac{3 y}{4 x ^{2}}

Calculate

x^{2} + (y 2 x) 2 = 1

Simplify the expression

x^{2} + 2 y 2 x = 1

Take the derivative of both sides

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

Calculate the derivative

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\frac{2 x 2 x + 4 x \frac{d y}{d x} + 2 y}{2 x} = \frac{d}{d x} (1)

Calculate the derivative

\frac{2 x 2 x + 4 x \frac{d y}{d x} + 2 y}{2 x} = 0

Rewrite the expression

\frac{2 x 2 x + 2 y + 4 x \frac{d y}{d x}}{2 x} = 0

Simplify

2 x 2 x + 2 y + 4 x \frac{d y}{d x} = 0

Move the constant to the right side

4 x \frac{d y}{d x} = 0 - (2 x 2 x + 2 y)

Subtract the terms

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

Divide both sides

\frac{4 x \frac{d y}{d x}}{4 x} = \frac{- 2 x 2 x - 2 y}{4 x}

Divide the numbers

\frac{d y}{d x} = \frac{- 2 x 2 x - 2 y}{4 x}

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

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

Calculate

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

Calculate

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

Calculate

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

Calculate

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\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{2 x ^{2} + 2 \frac{d y}{d x} \times 2 x \times x - 2 y 2 x}{2 x}}{4 x ^{2}}

Calculate

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

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

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

Solution

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\frac{d ^{2} y}{d x ^{2}} = \frac{3 y}{4 x ^{2}}

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