Solve (y - 1)^2 = x

Identify the conic

Find the vertex of the parabola

Find the focus of the parabola

Find the directrix of the parabola

(0, 1)

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Solve for x

Solve for y

x = y^{2} - 2 y + 1

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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}{2 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}{4 y ^{3} - 12 y ^{2} + 12 y - 4}

Calculate

(y - 1)^{2} = x

Take the derivative of both sides

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

Calculate the derivative

More Steps

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

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

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

Rewrite the expression

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

Divide both sides

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Rewrite the expression in exponential form

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

Calculate the derivative

More Steps

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

Rewrite the expression

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

Calculate

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

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

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{1}{4 y ^{3} - 12 y ^{2} + 12 y - 4}

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r = \frac{2 sin ( θ ) + cos ( θ ) + 2 sin ( 2 θ ) + cos ^{2} ( θ )}{2 sin ^{2} ( θ )} r = \frac{2 sin ( θ ) + cos ( θ ) - 2 sin ( 2 θ ) + cos ^{2} ( θ )}{2 sin ^{2} ( θ )}

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