Solve 7x^21/2y^4=3

Question

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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r = \frac{6 6}{6 7 cos ^{2} ( θ ) sin ^{4} ( θ )} r = - \frac{6 6}{6 7 cos ^{2} ( θ ) sin ^{4} ( 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{y}{2 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{3 y}{4 x ^{2}}

Calculate

7 x^{2} \frac{1}{2} y^{4} = 3

Simplify the expression

\frac{7}{2} x^{2} y^{4} = 3

Take the derivative of both sides

\frac{d}{d x} (\frac{7}{2} x^{2} y^{4}) = \frac{d}{d x} (3)

Calculate the derivative

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7 x y^{4} + 14 x^{2} y^{3} \frac{d y}{d x} = \frac{d}{d x} (3)

Calculate the derivative

7 x y^{4} + 14 x^{2} y^{3} \frac{d y}{d x} = 0

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

14 x^{2} y^{3} \frac{d y}{d x} = 0 - 7 x y^{4}

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

14 x^{2} y^{3} \frac{d y}{d x} = - 7 x y^{4}

Divide both sides

\frac{14 x ^{2} y ^{3} \frac{d y}{d x}}{14 x ^{2} y ^{3}} = \frac{- 7 x y ^{4}}{14 x ^{2} y ^{3}}

Divide the numbers

\frac{d y}{d x} = \frac{- 7 x y ^{4}}{14 x ^{2} y ^{3}}

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Use the commutative property to reorder the terms

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

Use the commutative property to reorder the terms

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

Calculate

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

Calculate

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

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

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

Solution

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

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