Solve 6x-2y^5=x^2

Question

Solve the equation

Solve for x

Solve for y

x = 3 + 9 - 2 y^{5} x = 3 - 9 - 2 y^{5}

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Testing for symmetry

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 first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{- x + 3}{5 y ^{4}}

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Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = \frac{- 5 y ^{5} - 4 x ^{2} + 24 x - 36}{25 y ^{9}}

Calculate

6 x - 2 y^{5} = x^{2}

Take the derivative of both sides

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

Calculate the derivative

More Steps

6 - 10 y^{4} \frac{d y}{d x} = \frac{d}{d x} (x^{2})

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

6 - 10 y^{4} \frac{d y}{d x} = 2 x

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

- 10 y^{4} \frac{d y}{d x} = 2 x - 6

Divide both sides

\frac{- 10 y ^{4} \frac{d y}{d x}}{- 10 y ^{4}} = \frac{2 x - 6}{- 10 y ^{4}}

Divide the numbers

\frac{d y}{d x} = \frac{2 x - 6}{- 10 y ^{4}}

Divide the numbers

More Steps

\frac{d y}{d x} = \frac{- x + 3}{5 y ^{4}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{- x + 3}{5 y ^{4}})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{- x + 3}{5 y ^{4}})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( - x + 3 ) \times 5 y ^{4} - ( - x + 3 ) \times \frac{d}{d x} ( 5 y ^{4} )}{( 5 y ^{4} ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 5 y ^{4} - ( - x + 3 ) \times \frac{d}{d x} ( 5 y ^{4} )}{( 5 y ^{4} ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 5 y ^{4} - ( - x + 3 ) \times 20 y ^{3} \frac{d y}{d x}}{( 5 y ^{4} ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 5 y ^{4} - ( - 20 x y ^{3} \frac{d y}{d x} + 60 y ^{3} \frac{d y}{d x} )}{( 5 y ^{4} ) ^{2}}

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

\frac{d ^{2} y}{d x ^{2}} = \frac{- 5 y ^{4} + 20 x y ^{3} \frac{d y}{d x} - 60 y ^{3} \frac{d y}{d x}}{( 5 y ^{4} ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 5 y ^{4} + 20 x y ^{3} \frac{d y}{d x} - 60 y ^{3} \frac{d y}{d x}}{25 y ^{8}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{- y + 4 x \frac{d y}{d x} - 12 \frac{d y}{d x}}{5 y ^{5}}

Use equation \frac{d y}{d x} = \frac{- x + 3}{5 y ^{4}} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{- y + 4 x \times \frac{- x + 3}{5 y ^{4}} - 12 \times \frac{- x + 3}{5 y ^{4}}}{5 y ^{5}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 5 y ^{5} - 4 x ^{2} + 24 x - 36}{25 y ^{9}}

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