Solve y^3y-6x=24 - ScanMath

Solve the equation

Solve for x

Solve for y

x = \frac{- 24 + y ^{4}}{6}

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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{3}{2 y ^{3}}

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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{27}{4 y ^{7}}

Calculate

y^{3} y - 6 x = 24

Simplify the expression

y^{4} - 6 x = 24

Take the derivative of both sides

\frac{d}{d x} (y^{4} - 6 x) = \frac{d}{d x} (24)

Calculate the derivative

More Steps

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

Calculate the derivative

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

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

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

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

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

Divide both sides

\frac{4 y ^{3} \frac{d y}{d x}}{4 y ^{3}} = \frac{6}{4 y ^{3}}

Divide the numbers

\frac{d y}{d x} = \frac{6}{4 y ^{3}}

Cancel out the common factor 2

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Rewrite the expression in exponential form

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

Calculate the derivative

More Steps

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

Rewrite the expression

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

Calculate

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

Use equation \frac{d y}{d x} = \frac{3}{2 y ^{3}} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{9 \times \frac{3}{2 y ^{3}}}{2 y ^{4}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{27}{4 y ^{7}}

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