Solve 3(x^5-y) 3x=12

Question

Solve the equation

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

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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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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{6 x ^{5} - y}{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{18 x ^{5} + 2 y}{x ^{2}}

Calculate

3 (x^{5} - y) 3 x = 12

Simplify the expression

9 x (x^{5} - y) = 12

Take the derivative of both sides

\frac{d}{d x} (9 x (x^{5} - y)) = \frac{d}{d x} (12)

Calculate the derivative

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9 x^{5} - 9 y + 45 x^{5} - 9 x \frac{d y}{d x} = \frac{d}{d x} (12)

Calculate the derivative

9 x^{5} - 9 y + 45 x^{5} - 9 x \frac{d y}{d x} = 0

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

- 9 x \frac{d y}{d x} = 0 - (9 x^{5} - 9 y + 45 x^{5})

Subtract the terms

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- 9 x \frac{d y}{d x} = - 54 x^{5} + 9 y

Divide both sides

\frac{- 9 x \frac{d y}{d x}}{- 9 x} = \frac{- 54 x ^{5} + 9 y}{- 9 x}

Divide the numbers

\frac{d y}{d x} = \frac{- 54 x ^{5} + 9 y}{- 9 x}

Divide the numbers

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\frac{d y}{d x} = \frac{6 x ^{5} - y}{x}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{6 x ^{5} - y}{x})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{6 x ^{5} - y}{x})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 6 x ^{5} - y ) \times x - ( 6 x ^{5} - y ) \times \frac{d}{d x} ( x )}{x ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{( 30 x ^{4} - \frac{d y}{d x} ) x - ( 6 x ^{5} - y ) \times \frac{d}{d x} ( x )}{x ^{2}}

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{( 30 x ^{4} - \frac{d y}{d x} ) x - ( 6 x ^{5} - y ) \times 1}{x ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{30 x ^{5} - x \frac{d y}{d x} - ( 6 x ^{5} - y ) \times 1}{x ^{2}}

Any expression multiplied by 1 remains the same

\frac{d ^{2} y}{d x ^{2}} = \frac{30 x ^{5} - x \frac{d y}{d x} - ( 6 x ^{5} - y )}{x ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{24 x ^{5} - x \frac{d y}{d x} + y}{x ^{2}}

Use equation \frac{d y}{d x} = \frac{6 x ^{5} - y}{x} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{24 x ^{5} - x \times \frac{6 x ^{5} - y}{x} + y}{x ^{2}}

Solution

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\frac{d ^{2} y}{d x ^{2}} = \frac{18 x ^{5} + 2 y}{x ^{2}}

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