Solve y^2 = 4 - 2x

Identify the conic

Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

y^{2} = - 2 (x - 2)

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Solve for x

Solve for y

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

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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}{y}

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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}{y ^{3}}

Calculate

y^{2} = 4 - 2 x

Take the derivative of both sides

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

Calculate the derivative

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

Calculate the derivative

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Rewrite the expression in exponential form

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

Calculate the derivative

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

Rewrite the expression

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

Calculate

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

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

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

Solution

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

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r = \frac{- cos ( θ ) + 1 + 3 sin ^{2} ( θ )}{sin ^{2} ( θ )} r = - \frac{cos ( θ ) + 1 + 3 sin ^{2} ( θ )}{sin ^{2} ( θ )}

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