Solve x - 4y^5 = 0

Question

Solve the equation

Solve for x

Solve for y

x = 4 y^{5}

Evaluate

x - 4 y^{5} = 0

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

x = 0 + 4 y^{5}

Solution

x = 4 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

Symmetry with respect to the origin

Evaluate

x - 4 y^{5} = 0

To test if the graph of x - 4 y^{5} = 0 is symmetry with respect to the origin,substitute -x for x and -y for y

- x - 4 (- y)^{5} = 0

Evaluate

More Steps

Evaluate

- x - 4 (- y)^{5}

Multiply the terms

More Steps

Evaluate

4 (- y)^{5}

Rewrite the expression

4 (- y^{5})

Multiply the numbers

- 4 y^{5}

- x - (- 4 y^{5})

Rewrite the expression

- x + 4 y^{5}

- x + 4 y^{5} = 0

Solution

Symmetry with respect to the origin

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Rewrite the equation

r = 0 r = 4 \frac{cos ( θ )}{4 sin ^{5} ( θ )} r = - 4 \frac{cos ( θ )}{4 sin ^{5} ( θ )}

Evaluate

x - 4 y^{5} = 0

To convert the equation to polar coordinates,substitute x for r cos (θ) and y for r sin (θ)

cos (θ) \times r - 4 (sin (θ) \times r)^{5} = 0

Factor the expression

- 4 sin^{5} (θ) \times r^{5} + cos (θ) \times r = 0

Factor the expression

r (- 4 sin^{5} (θ) \times r^{4} + cos (θ)) = 0

When the product of factors equals 0,at least one factor is 0

r = 0 - 4 sin^{5} (θ) \times r^{4} + cos (θ) = 0

Solution

More Steps

Factor the expression

- 4 sin^{5} (θ) \times r^{4} + cos (θ) = 0

Subtract the terms

- 4 sin^{5} (θ) \times r^{4} + cos (θ) - cos (θ) = 0 - cos (θ)

Evaluate

- 4 sin^{5} (θ) \times r^{4} = - cos (θ)

Divide the terms

r^{4} = \frac{cos ( θ )}{4 sin ^{5} ( θ )}

Evaluate the power

r = \pm 4 \frac{cos ( θ )}{4 sin ^{5} ( θ )}

Separate into possible cases

r = 4 \frac{cos ( θ )}{4 sin ^{5} ( θ )} r = - 4 \frac{cos ( θ )}{4 sin ^{5} ( θ )}

r = 0 r = 4 \frac{cos ( θ )}{4 sin ^{5} ( θ )} r = - 4 \frac{cos ( θ )}{4 sin ^{5} ( θ )}

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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{1}{20 y ^{4}}

Calculate

x - 4 y^{5} = 0

Take the derivative of both sides

\frac{d}{d x} (x - 4 y^{5}) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

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

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

Evaluate the derivative

More Steps

Evaluate

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

Use differentiation rules

\frac{d}{d y} (- 4 y^{5}) \times \frac{d y}{d x}

Evaluate the derivative

- 20 y^{4} \frac{d y}{d x}

1 - 20 y^{4} \frac{d y}{d x}

1 - 20 y^{4} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

1 - 20 y^{4} \frac{d y}{d x} = 0

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

- 20 y^{4} \frac{d y}{d x} = 0 - 1

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

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

Divide both sides

\frac{- 20 y ^{4} \frac{d y}{d x}}{- 20 y ^{4}} = \frac{- 1}{- 20 y ^{4}}

Divide the numbers

\frac{d y}{d x} = \frac{- 1}{- 20 y ^{4}}

Solution

\frac{d y}{d x} = \frac{1}{20 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{1}{100 y ^{9}}

Calculate

x - 4 y^{5} = 0

Take the derivative of both sides

\frac{d}{d x} (x - 4 y^{5}) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

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

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

Evaluate the derivative

More Steps

Evaluate

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

Use differentiation rules

\frac{d}{d y} (- 4 y^{5}) \times \frac{d y}{d x}

Evaluate the derivative

- 20 y^{4} \frac{d y}{d x}

1 - 20 y^{4} \frac{d y}{d x}

1 - 20 y^{4} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

1 - 20 y^{4} \frac{d y}{d x} = 0

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

- 20 y^{4} \frac{d y}{d x} = 0 - 1

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

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

Divide both sides

\frac{- 20 y ^{4} \frac{d y}{d x}}{- 20 y ^{4}} = \frac{- 1}{- 20 y ^{4}}

Divide the numbers

\frac{d y}{d x} = \frac{- 1}{- 20 y ^{4}}

Cancel out the common factor - 1

\frac{d y}{d x} = \frac{1}{20 y ^{4}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{1}{20 y ^{4}})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{1}{20 y ^{4}})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{1}{20} \times \frac{d}{d x} (\frac{1}{y ^{4}})

Rewrite the expression in exponential form

\frac{d ^{2} y}{d x ^{2}} = \frac{1}{20} \times \frac{d}{d x} (y^{- 4})

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

\frac{d}{d y} (y^{- 4}) \times \frac{d y}{d x}

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

- 4 y^{- 5} \frac{d y}{d x}

\frac{d ^{2} y}{d x ^{2}} = \frac{1}{20} (- 4 y^{- 5} \frac{d y}{d x})

Rewrite the expression

\frac{d ^{2} y}{d x ^{2}} = \frac{1}{20} (- \frac{4 \frac{d y}{d x}}{y ^{5}})

Calculate

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

Use equation \frac{d y}{d x} = \frac{1}{20 y ^{4}} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{1}{20 y ^{4}}}{5 y ^{5}}

Solution

More Steps

Calculate

- \frac{\frac{1}{20 y ^{4}}}{5 y ^{5}}

Divide the terms

More Steps

Evaluate

\frac{\frac{1}{20 y ^{4}}}{5 y ^{5}}

Multiply by the reciprocal

\frac{1}{20 y ^{4}} \times \frac{1}{5 y ^{5}}

Multiply the terms

\frac{1}{20 y ^{4} \times 5 y ^{5}}

Multiply the terms

\frac{1}{100 y ^{9}}

- \frac{1}{100 y ^{9}}

\frac{d ^{2} y}{d x ^{2}} = - \frac{1}{100 y ^{9}}

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