Solve y^2 - 8 x = 0

Identify the conic

Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

y^{2} = 8 x

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Solve for x

Solve for y

x = \frac{y ^{2}}{8}

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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{4}{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{16}{y ^{3}}

Calculate

y^{2} - 8 x = 0

Take the derivative of both sides

\frac{d}{d x} (y^{2} - 8 x) = \frac{d}{d x} (0)

Calculate the derivative

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2 y \frac{d y}{d x} - 8 = \frac{d}{d x} (0)

Calculate the derivative

2 y \frac{d y}{d x} - 8 = 0

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

2 y \frac{d y}{d x} = 0 + 8

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

2 y \frac{d y}{d x} = 8

Divide both sides

\frac{2 y \frac{d y}{d x}}{2 y} = \frac{8}{2 y}

Divide the numbers

\frac{d y}{d x} = \frac{8}{2 y}

Cancel out the common factor 2

\frac{d y}{d x} = \frac{4}{y}

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Rewrite the expression in exponential form

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

Calculate the derivative

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

Rewrite the expression

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

Calculate

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

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

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

Solution

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\frac{d ^{2} y}{d x ^{2}} = - \frac{16}{y ^{3}}

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r = 0 r = 8 cos (θ) csc^{2} (θ)

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