Solve 6y^3x = 24 - ScanMath

Question

Solve the equation

Solve for x

Solve for y

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

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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 = 4 4 csc^{3} (θ) sec (θ) r = - 4 4 csc^{3} (θ) sec (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}{3 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{4 y}{9 x ^{2}}

Calculate

6 y^{3} x = 24

Take the derivative of both sides

\frac{d}{d x} (6 y^{3} x) = \frac{d}{d x} (24)

Calculate the derivative

More Steps

6 y^{3} + 18 x y^{2} \frac{d y}{d x} = \frac{d}{d x} (24)

Calculate the derivative

6 y^{3} + 18 x y^{2} \frac{d y}{d x} = 0

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

18 x y^{2} \frac{d y}{d x} = 0 - 6 y^{3}

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

18 x y^{2} \frac{d y}{d x} = - 6 y^{3}

Divide both sides

\frac{18 x y ^{2} \frac{d y}{d x}}{18 x y ^{2}} = \frac{- 6 y ^{3}}{18 x y ^{2}}

Divide the numbers

\frac{d y}{d x} = \frac{- 6 y ^{3}}{18 x y ^{2}}

Divide the numbers

More Steps

\frac{d y}{d x} = - \frac{y}{3 x}

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

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

Use the commutative property to reorder the terms

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

Use the commutative property to reorder the terms

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

Use the commutative property to reorder the terms

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

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{3 x \frac{d y}{d x} - 3 y}{9 x ^{2}}

Calculate

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

Use equation \frac{d y}{d x} = - \frac{y}{3 x} to substitute

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{4 y}{9 x ^{2}}

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