Solve 48x-72y=30y^24

Identify the conic

Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

(y + \frac{3}{10})^{2} = \frac{2}{5} (x + \frac{9}{40})

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Solve the equation

Solve for x

Solve for y

x = \frac{5 y ^{2} + 3 y}{2}

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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{2}{3 + 10 y}

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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{40}{27 + 270 y + 900 y ^{2} + 1000 y ^{3}}

Calculate

48 x - 72 y = 30 y^{2} 4

Simplify the expression

48 x - 72 y = 120 y^{2}

Take the derivative of both sides

\frac{d}{d x} (48 x - 72 y) = \frac{d}{d x} (120 y^{2})

Calculate the derivative

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48 - 72 \frac{d y}{d x} = \frac{d}{d x} (120 y^{2})

Calculate the derivative

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48 - 72 \frac{d y}{d x} = 240 y \frac{d y}{d x}

Move the variable to the left side

48 - 72 \frac{d y}{d x} - 240 y \frac{d y}{d x} = 0

Collect like terms by calculating the sum or difference of their coefficients

48 + (- 72 - 240 y) \frac{d y}{d x} = 0

Move the constant to the right side

(- 72 - 240 y) \frac{d y}{d x} = 0 - 48

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

(- 72 - 240 y) \frac{d y}{d x} = - 48

Divide both sides

\frac{( - 72 - 240 y ) \frac{d y}{d x}}{- 72 - 240 y} = \frac{- 48}{- 72 - 240 y}

Divide the numbers

\frac{d y}{d x} = \frac{- 48}{- 72 - 240 y}

Divide the numbers

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\frac{d y}{d x} = \frac{2}{3 + 10 y}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{2}{3 + 10 y})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{2}{3 + 10 y})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = 2 \times \frac{d}{d x} (\frac{1}{3 + 10 y})

Rewrite the expression in exponential form

\frac{d ^{2} y}{d x ^{2}} = 2 \times \frac{d}{d x} ((3 + 10 y)^{- 1})

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = 2 (- 10 \frac{d y}{d x} \times (3 + 10 y)^{- 2})

Rewrite the expression

\frac{d ^{2} y}{d x ^{2}} = 2 (- \frac{10 \frac{d y}{d x}}{( 3 + 10 y ) ^{2}})

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{20 \frac{d y}{d x}}{( 3 + 10 y ) ^{2}}

Use equation \frac{d y}{d x} = \frac{2}{3 + 10 y} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{20 \times \frac{2}{3 + 10 y}}{( 3 + 10 y ) ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{40}{27 + 270 y + 900 y ^{2} + 1000 y ^{3}}

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Rewrite the equation

r = 0 r = \frac{2 cos ( θ ) csc ^{2} ( θ ) - 3 csc ( θ )}{5}

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Identify the conic

Solve the equation

Testing for symmetry

Find the first derivative

Find the second derivative

Rewrite the equation

Graph