Solve x y^2-10=0 - ScanMath

Question

Solve the equation

Solve for x

Solve for y

x = \frac{10}{y ^{2}}

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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 = 3 10 sec (θ) csc^{2} (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

x y^{2} - 10 = 0

Take the derivative of both sides

\frac{d}{d x} (x y^{2} - 10) = \frac{d}{d x} (0)

Calculate the derivative

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

Calculate the derivative

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

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

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

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

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

More Steps

\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

More Steps

\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

More Steps

\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

More Steps

\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

More Steps

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

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