Solve y^3=-1/4(x-8)

Solve the equation

Solve for x

Solve for y

x = - 4 y^{3} + 8

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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{1}{12 y ^{2}}

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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{1}{72 y ^{5}}

Calculate

y^{3} = - \frac{1}{4} (x - 8)

Simplify the expression

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

Take the derivative of both sides

\frac{d}{d x} (y^{3}) = \frac{d}{d x} (- \frac{1}{4} x + 2)

Calculate the derivative

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

Calculate the derivative

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

Multiply by the reciprocal

3 y^{2} \frac{d y}{d x} \times \frac{1}{3 y ^{2}} = - \frac{1}{4} \times \frac{1}{3 y ^{2}}

Multiply

\frac{d y}{d x} = - \frac{1}{4} \times \frac{1}{3 y ^{2}}

Multiply

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\frac{d y}{d x} = - \frac{1}{12 y ^{2}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{1}{12 y ^{2}})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{1}{12 y ^{2}})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{1}{12} \times \frac{d}{d x} (\frac{1}{y ^{2}})

Rewrite the expression in exponential form

\frac{d ^{2} y}{d x ^{2}} = - \frac{1}{12} \times \frac{d}{d x} (y^{- 2})

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{1}{12} (- 2 y^{- 3} \frac{d y}{d x})

Rewrite the expression

\frac{d ^{2} y}{d x ^{2}} = - \frac{1}{12} (- \frac{2 \frac{d y}{d x}}{y ^{3}})

Calculate

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

Use equation \frac{d y}{d x} = - \frac{1}{12 y ^{2}} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{- \frac{1}{12 y ^{2}}}{6 y ^{3}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{1}{72 y ^{5}}

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