Find the foci of the ellipse of \frac{x^2}{36}+\frac{y^2}{9} = 1

F_{1} = \left(-3\sqrt{3},0\right), F_{2} = \left(3\sqrt{3},0\right)

Find the vertices of the ellipse of \frac{x^2}{36}+\frac{y^2}{9} = 1

V_{1} = \left(0,3\right), V_{2} = \left(0,-3\right), V_{3} = \left(6,0\right), V_{4} = \left(-6,0\right)

\text{Solve for }x of \frac{x^2}{36}+\frac{y^2}{9} = 1

\begin{align}&x=2\sqrt{9-y^{2}}\\&x=-2\sqrt{9-y^{2}}\end{align}

\text{Solve for }y of \frac{x^2}{36}+\frac{y^2}{9} = 1

\begin{align}&y=\frac{\sqrt{36-x^{2}}}{2}\\&y=-\frac{\sqrt{36-x^{2}}}{2}\end{align}

Testing for symmetry about the origin of \frac{x^2}{36}+\frac{y^2}{9} = 1

\textrm{Symmetry with respect to the origin}

Testing for symmetry about the x-axis of \frac{x^2}{36}+\frac{y^2}{9} = 1

\textrm{Symmetry with respect to the x-axis}

Testing for symmetry about the y-axis of \frac{x^2}{36}+\frac{y^2}{9} = 1

\textrm{Symmetry with respect to the y-axis}

\text{Find the second derivative with respect to }y of \frac{x^2}{36}+\frac{y^2}{9} = 1

\frac{d^2x}{dy^2}=-\frac{4x^{2}+16y^{2}}{x^{3}}

Rewrite in polar form of \frac{x^2}{36}+\frac{y^2}{9} = 1

\begin{align}&r=\frac{6\sqrt{1+3\sin^{2}\left(\theta \right)}}{1+3\sin^{2}\left(\theta \right)}\\&r=-\frac{6\sqrt{1+3\sin^{2}\left(\theta \right)}}{1+3\sin^{2}\left(\theta \right)}\end{align}

Solve fracx^236+fracy^29 = 1

Q: \text{Find the second derivative with respect to }x of \frac{x^2}{36}+\frac{y^2}{9} = 1

\frac{d^2y}{dx^2}=-\frac{4y^{2}+x^{2}}{16y^{3}}

Identify the conic

Find the center of the ellipse

Find the foci of the ellipse

Find the vertices of the ellipse

(0, 0)

Show Solution

Solve the equation

Solve for x

Solve for y

x = 2 9 - y^{2} x = - 2 9 - y^{2}

Show Solution

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

Show Solution

Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

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

Show Solution

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{4 y ^{2} + x ^{2}}{16 y ^{3}}

Calculate

\frac{x ^{2}}{36} + \frac{y ^{2}}{9} = 1

Take the derivative of both sides

\frac{d}{d x} (\frac{x ^{2}}{36} + \frac{y ^{2}}{9}) = \frac{d}{d x} (1)

Calculate the derivative

More Steps

\frac{x + 4 y \frac{d y}{d x}}{18} = \frac{d}{d x} (1)

Calculate the derivative

\frac{x + 4 y \frac{d y}{d x}}{18} = 0

Simplify

x + 4 y \frac{d y}{d x} = 0

Move the constant to the right side

4 y \frac{d y}{d x} = 0 - x

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

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

Divide both sides

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

Divide the numbers

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

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

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

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

Calculate the derivative

More Steps

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

Any expression multiplied by 1 remains the same

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

Use the commutative property to reorder the terms

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

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y - 4 x \frac{d y}{d x}}{16 y ^{2}}

Calculate

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

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

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y ^{2} + x ^{2}}{16 y ^{3}}

Show Solution

Rewrite the equation

r = \frac{6 1 + 3 sin ^{2} ( θ )}{1 + 3 sin ^{2} ( θ )} r = - \frac{6 1 + 3 sin ^{2} ( θ )}{1 + 3 sin ^{2} ( θ )}

Show Solution

fracx^236+fracy^29 = 1

Identify the conic

Solve the equation

Testing for symmetry

Find the first derivative

Find the second derivative

Rewrite the equation

Graph

Frequently Asked Questions

Find the center of the ellipse of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Find the foci of the ellipse of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Find the vertices of the ellipse of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Find the eccentricity of the ellipse of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

$Solve for x$ of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

$Solve for y$ of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Testing for symmetry about the origin of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Testing for symmetry about the x-axis of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Testing for symmetry about the y-axis of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

$Find the derivative with respect to x$ of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

$Find the derivative with respect to y$ of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

$Find the second derivative with respect to x$ of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

$Find the second derivative with respect to y$ of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Rewrite in polar form of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

fracx^236+fracy^29 = 1

Identify the conic

Solve the equation

Testing for symmetry

Find the first derivative

Find the second derivative

Rewrite the equation

Graph

Frequently Asked Questions

Find the center of the ellipse of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Find the foci of the ellipse of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Find the vertices of the ellipse of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Find the eccentricity of the ellipse of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Solve for x of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Solve for y of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Testing for symmetry about the origin of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Testing for symmetry about the x-axis of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Testing for symmetry about the y-axis of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Find the derivative with respect to x of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Find the derivative with respect to y of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Find the second derivative with respect to x of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Find the second derivative with respect to y of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

Rewrite in polar form of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

$Solve for x$ of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

$Solve for y$ of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

$Find the derivative with respect to x$ of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

$Find the derivative with respect to y$ of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

$Find the second derivative with respect to x$ of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)

$Find the second derivative with respect to y$ of \(\frac{x^2}{36}+\frac{y^2}{9} = 1\)