Calculus Examples

$x^{2} y^{2} - 9 x^{2} - 4 y^{2} = 0$

Step 1

Solve the equation as $y$ in terms of $x$ .

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Step 1.1

Add $9 x^{2}$ to both sides of the equation.

$x^{2} y^{2} - 4 y^{2} = 9 x^{2}$

Step 1.2

Factor $y^{2}$ out of $x^{2} y^{2} - 4 y^{2}$ .

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Step 1.2.1

Factor $y^{2}$ out of $x^{2} y^{2}$ .

$y^{2} x^{2} - 4 y^{2} = 9 x^{2}$

Step 1.2.2

Factor $y^{2}$ out of $- 4 y^{2}$ .

$y^{2} x^{2} + y^{2} \cdot - 4 = 9 x^{2}$

Step 1.2.3

Factor $y^{2}$ out of $y^{2} x^{2} + y^{2} \cdot - 4$ .

$y^{2} (x^{2} - 4) = 9 x^{2}$

Step 1.3

Rewrite $4$ as $2^{2}$ .

$y^{2} (x^{2} - 2^{2}) = 9 x^{2}$

Step 1.4

Factor.

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Step 1.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$y^{2} ((x + 2) (x - 2)) = 9 x^{2}$

Step 1.4.2

Remove unnecessary parentheses.

$y^{2} (x + 2) (x - 2) = 9 x^{2}$

Step 1.5

Divide each term in $y^{2} (x + 2) (x - 2) = 9 x^{2}$ by $(x + 2) (x - 2)$ and simplify.

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Step 1.5.1

Divide each term in $y^{2} (x + 2) (x - 2) = 9 x^{2}$ by $(x + 2) (x - 2)$ .

$\frac{y^{2} (x + 2) (x - 2)}{(x + 2) (x - 2)} = \frac{9 x^{2}}{(x + 2) (x - 2)}$

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Cancel the common factor of $x + 2$ .

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Step 1.5.2.1.1

Cancel the common factor.

$\frac{y^{2} (x + 2) (x - 2)}{(x + 2) (x - 2)} = \frac{9 x^{2}}{(x + 2) (x - 2)}$

Step 1.5.2.1.2

Rewrite the expression.

$\frac{y^{2} (x - 2)}{x - 2} = \frac{9 x^{2}}{(x + 2) (x - 2)}$

Step 1.5.2.2

Cancel the common factor of $x - 2$ .

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Step 1.5.2.2.1

Cancel the common factor.

$\frac{y^{2} (x - 2)}{x - 2} = \frac{9 x^{2}}{(x + 2) (x - 2)}$

Step 1.5.2.2.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{9 x^{2}}{(x + 2) (x - 2)}$

Step 1.6

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{\frac{9 x^{2}}{(x + 2) (x - 2)}}$

Step 1.7

Simplify $\pm \sqrt{\frac{9 x^{2}}{(x + 2) (x - 2)}}$ .

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Step 1.7.1

Rewrite $\sqrt{\frac{9 x^{2}}{(x + 2) (x - 2)}}$ as $\frac{\sqrt{9 x^{2}}}{\sqrt{(x + 2) (x - 2)}}$ .

$y = \pm \frac{\sqrt{9 x^{2}}}{\sqrt{(x + 2) (x - 2)}}$

Step 1.7.2

Simplify the numerator.

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Step 1.7.2.1

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$y = \pm \frac{\sqrt{{(3 x)}^{2}}}{\sqrt{(x + 2) (x - 2)}}$

Step 1.7.2.2

Pull terms out from under the radical, assuming positive real numbers.

$y = \pm \frac{3 x}{\sqrt{(x + 2) (x - 2)}}$

Step 1.7.3

Multiply $\frac{3 x}{\sqrt{(x + 2) (x - 2)}}$ by $\frac{\sqrt{(x + 2) (x - 2)}}{\sqrt{(x + 2) (x - 2)}}$ .

$y = \pm \frac{3 x}{\sqrt{(x + 2) (x - 2)}} \cdot \frac{\sqrt{(x + 2) (x - 2)}}{\sqrt{(x + 2) (x - 2)}}$

Step 1.7.4

Combine and simplify the denominator.

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Step 1.7.4.1

Multiply $\frac{3 x}{\sqrt{(x + 2) (x - 2)}}$ by $\frac{\sqrt{(x + 2) (x - 2)}}{\sqrt{(x + 2) (x - 2)}}$ .

$y = \pm \frac{3 x \sqrt{(x + 2) (x - 2)}}{\sqrt{(x + 2) (x - 2)} \sqrt{(x + 2) (x - 2)}}$

Step 1.7.4.2

Raise $\sqrt{(x + 2) (x - 2)}$ to the power of $1$ .

$y = \pm \frac{3 x \sqrt{(x + 2) (x - 2)}}{{\sqrt{(x + 2) (x - 2)}}^{1} \sqrt{(x + 2) (x - 2)}}$

Step 1.7.4.3

Raise $\sqrt{(x + 2) (x - 2)}$ to the power of $1$ .

$y = \pm \frac{3 x \sqrt{(x + 2) (x - 2)}}{{\sqrt{(x + 2) (x - 2)}}^{1} {\sqrt{(x + 2) (x - 2)}}^{1}}$

Step 1.7.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{3 x \sqrt{(x + 2) (x - 2)}}{{\sqrt{(x + 2) (x - 2)}}^{1 + 1}}$

Step 1.7.4.5

Add $1$ and $1$ .

$y = \pm \frac{3 x \sqrt{(x + 2) (x - 2)}}{{\sqrt{(x + 2) (x - 2)}}^{2}}$

Step 1.7.4.6

Rewrite ${\sqrt{(x + 2) (x - 2)}}^{2}$ as $(x + 2) (x - 2)$ .

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Step 1.7.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x + 2) (x - 2)}$ as ${((x + 2) (x - 2))}^{\frac{1}{2}}$ .

$y = \pm \frac{3 x \sqrt{(x + 2) (x - 2)}}{{({((x + 2) (x - 2))}^{\frac{1}{2}})}^{2}}$

Step 1.7.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm \frac{3 x \sqrt{(x + 2) (x - 2)}}{{((x + 2) (x - 2))}^{\frac{1}{2} \cdot 2}}$

Step 1.7.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \pm \frac{3 x \sqrt{(x + 2) (x - 2)}}{{((x + 2) (x - 2))}^{\frac{2}{2}}}$

Step 1.7.4.6.4

Cancel the common factor of $2$ .

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Step 1.7.4.6.4.1

Cancel the common factor.

$y = \pm \frac{3 x \sqrt{(x + 2) (x - 2)}}{{((x + 2) (x - 2))}^{\frac{2}{2}}}$

Step 1.7.4.6.4.2

Rewrite the expression.

$y = \pm \frac{3 x \sqrt{(x + 2) (x - 2)}}{{((x + 2) (x - 2))}^{1}}$

Step 1.7.4.6.5

Simplify.

$y = \pm \frac{3 x \sqrt{(x + 2) (x - 2)}}{(x + 2) (x - 2)}$

Step 1.8

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.8.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{3 x \sqrt{(x + 2) (x - 2)}}{(x + 2) (x - 2)}$

Step 1.8.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{3 x \sqrt{(x + 2) (x - 2)}}{(x + 2) (x - 2)}$

Step 1.8.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{3 x \sqrt{(x + 2) (x - 2)}}{(x + 2) (x - 2)}$

$y = - \frac{3 x \sqrt{(x + 2) (x - 2)}}{(x + 2) (x - 2)}$

$y = \frac{3 x \sqrt{(x + 2) (x - 2)}}{(x + 2) (x - 2)}$

$y = - \frac{3 x \sqrt{(x + 2) (x - 2)}}{(x + 2) (x - 2)}$

$y = \frac{3 x \sqrt{(x + 2) (x - 2)}}{(x + 2) (x - 2)}$

$y = - \frac{3 x \sqrt{(x + 2) (x - 2)}}{(x + 2) (x - 2)}$

Step 2

Set each solution of $y$ as a function of $x$ .

$y = \frac{3 x \sqrt{(x + 2) (x - 2)}}{(x + 2) (x - 2)} \to f (x) = \frac{3 x \sqrt{(x + 2) (x - 2)}}{(x + 2) (x - 2)}$

$y = - \frac{3 x \sqrt{(x + 2) (x - 2)}}{(x + 2) (x - 2)} \to f (x) = - \frac{3 x \sqrt{(x + 2) (x - 2)}}{(x + 2) (x - 2)}$

Step 3

Because the $y$ variable in the equation $x^{2} y^{2} - 9 x^{2} - 4 y^{2} = 0$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

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Step 3.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} y^{2} - 9 x^{2} - 4 y^{2}) = \frac{d}{d x} (0)$

Step 3.2

Differentiate the left side of the equation.

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Step 3.2.1

By the Sum Rule, the derivative of $x^{2} y^{2} - 9 x^{2} - 4 y^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2} y^{2}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$ .

$\frac{d}{d x} [x^{2} y^{2}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 3.2.2

Evaluate $\frac{d}{d x} [x^{2} y^{2}]$ .

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Step 3.2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = y^{2}$ .

$x^{2} \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 3.2.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 3.2.2.2.1

To apply the Chain Rule, set $u_{1}$ as $y$ .

$x^{2} (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 3.2.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$x^{2} (2 u_{1} \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 3.2.2.2.3

Replace all occurrences of $u_{1}$ with $y$ .

$x^{2} (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 3.2.2.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$x^{2} (2 y y') + y^{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 3.2.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{2} (2 y y') + y^{2} (2 x) + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 3.2.2.5

Move $2$ to the left of $x^{2}$ .

$2 \cdot x^{2} y y' + y^{2} (2 x) + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 3.2.2.6

Move $2$ to the left of $y^{2}$ .

$2 x^{2} y y' + 2 y^{2} x + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 3.2.3

Evaluate $\frac{d}{d x} [- 9 x^{2}]$ .

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Step 3.2.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{2}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{2}]$ .

$2 x^{2} y y' + 2 y^{2} x - 9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 y^{2}]$

Step 3.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x^{2} y y' + 2 y^{2} x - 9 (2 x) + \frac{d}{d x} [- 4 y^{2}]$

Step 3.2.3.3

Multiply $2$ by $- 9$ .

$2 x^{2} y y' + 2 y^{2} x - 18 x + \frac{d}{d x} [- 4 y^{2}]$

Step 3.2.4

Evaluate $\frac{d}{d x} [- 4 y^{2}]$ .

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Step 3.2.4.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 y^{2}$ with respect to $x$ is $- 4 \frac{d}{d x} [y^{2}]$ .

$2 x^{2} y y' + 2 y^{2} x - 18 x - 4 \frac{d}{d x} [y^{2}]$

Step 3.2.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 3.2.4.2.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$2 x^{2} y y' + 2 y^{2} x - 18 x - 4 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y])$

Step 3.2.4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$2 x^{2} y y' + 2 y^{2} x - 18 x - 4 (2 u_{2} \frac{d}{d x} [y])$

Step 3.2.4.2.3

Replace all occurrences of $u_{2}$ with $y$ .

$2 x^{2} y y' + 2 y^{2} x - 18 x - 4 (2 y \frac{d}{d x} [y])$

Step 3.2.4.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$2 x^{2} y y' + 2 y^{2} x - 18 x - 4 (2 y y')$

Step 3.2.4.4

Multiply $2$ by $- 4$ .

$2 x^{2} y y' + 2 y^{2} x - 18 x - 8 y y'$

Step 3.2.5

Reorder terms.

$2 x^{2} y y' + 2 y^{2} x - 8 y y' - 18 x$

Step 3.3

Since $0$ is constant with respect to $x$ , the derivative of $0$ with respect to $x$ is $0$ .

$0$

Step 3.4

Reform the equation by setting the left side equal to the right side.

$2 x^{2} y y' + 2 y^{2} x - 8 y y' - 18 x = 0$

Step 3.5

Solve for $y'$ .

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Step 3.5.1

Move all terms not containing $y'$ to the right side of the equation.

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Step 3.5.1.1

Subtract $2 y^{2} x$ from both sides of the equation.

$2 x^{2} y y' - 8 y y' - 18 x = - 2 y^{2} x$

Step 3.5.1.2

Add $18 x$ to both sides of the equation.

$2 x^{2} y y' - 8 y y' = - 2 y^{2} x + 18 x$

Step 3.5.2

Factor $2 y y'$ out of $2 x^{2} y y' - 8 y y'$ .

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Step 3.5.2.1

Factor $2 y y'$ out of $2 x^{2} y y'$ .

$2 y y' x^{2} - 8 y y' = - 2 y^{2} x + 18 x$

Step 3.5.2.2

Factor $2 y y'$ out of $- 8 y y'$ .

$2 y y' x^{2} + 2 y y' \cdot - 4 = - 2 y^{2} x + 18 x$

Step 3.5.2.3

Factor $2 y y'$ out of $2 y y' x^{2} + 2 y y' \cdot - 4$ .

$2 y y' (x^{2} - 4) = - 2 y^{2} x + 18 x$

Step 3.5.3

Rewrite $4$ as $2^{2}$ .

$2 y y' (x^{2} - 2^{2}) = - 2 y^{2} x + 18 x$

Step 3.5.4

Factor.

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Step 3.5.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$2 y y' ((x + 2) (x - 2)) = - 2 y^{2} x + 18 x$

Step 3.5.4.2

Remove unnecessary parentheses.

$2 y y' (x + 2) (x - 2) = - 2 y^{2} x + 18 x$

Step 3.5.5

Divide each term in $2 y y' (x + 2) (x - 2) = - 2 y^{2} x + 18 x$ by $2 y (x + 2) (x - 2)$ and simplify.

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Step 3.5.5.1

Divide each term in $2 y y' (x + 2) (x - 2) = - 2 y^{2} x + 18 x$ by $2 y (x + 2) (x - 2)$ .

$\frac{2 y y' (x + 2) (x - 2)}{2 y (x + 2) (x - 2)} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.2

Simplify the left side.

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Step 3.5.5.2.1

Cancel the common factor of $2$ .

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Step 3.5.5.2.1.1

Cancel the common factor.

$\frac{2 y y' (x + 2) (x - 2)}{2 y (x + 2) (x - 2)} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.2.1.2

Rewrite the expression.

$\frac{y y' (x + 2) (x - 2)}{(y (x + 2)) (x - 2)} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.2.2

Cancel the common factor of $y$ .

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Step 3.5.5.2.2.1

Cancel the common factor.

$\frac{y y' (x + 2) (x - 2)}{y (x + 2) (x - 2)} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.2.2.2

Rewrite the expression.

$\frac{y' (x + 2) (x - 2)}{(x + 2) (x - 2)} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.2.3

Cancel the common factor of $x + 2$ .

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Step 3.5.5.2.3.1

Cancel the common factor.

$\frac{y' (x + 2) (x - 2)}{(x + 2) (x - 2)} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.2.3.2

Rewrite the expression.

$\frac{y' (x - 2)}{x - 2} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.2.4

Cancel the common factor of $x - 2$ .

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Step 3.5.5.2.4.1

Cancel the common factor.

$\frac{y' (x - 2)}{x - 2} = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.2.4.2

Divide $y'$ by $1$ .

$y' = \frac{- 2 y^{2} x}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.3

Simplify the right side.

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Step 3.5.5.3.1

Simplify each term.

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Step 3.5.5.3.1.1

Cancel the common factor of $- 2$ and $2$ .

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Step 3.5.5.3.1.1.1

Factor $2$ out of $- 2 y^{2} x$ .

$y' = \frac{2 (- y^{2} x)}{2 y (x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.3.1.1.2

Cancel the common factors.

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Step 3.5.5.3.1.1.2.1

Factor $2$ out of $2 y (x + 2) (x - 2)$ .

$y' = \frac{2 (- y^{2} x)}{2 ((y (x + 2)) (x - 2))} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.3.1.1.2.2

Cancel the common factor.

$y' = \frac{2 (- y^{2} x)}{2 ((y (x + 2)) (x - 2))} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.3.1.1.2.3

Rewrite the expression.

$y' = \frac{- y^{2} x}{(y (x + 2)) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.3.1.2

Cancel the common factor of $y^{2}$ and $y$ .

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Step 3.5.5.3.1.2.1

Factor $y$ out of $- y^{2} x$ .

$y' = \frac{y (- y x)}{(y (x + 2)) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.3.1.2.2

Cancel the common factors.

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Step 3.5.5.3.1.2.2.1

Factor $y$ out of $(y (x + 2)) (x - 2)$ .

$y' = \frac{y (- y x)}{y ((x + 2) (x - 2))} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.3.1.2.2.2

Cancel the common factor.

$y' = \frac{y (- y x)}{y ((x + 2) (x - 2))} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.3.1.2.2.3

Rewrite the expression.

$y' = \frac{- y x}{(x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.3.1.3

Move the negative in front of the fraction.

$y' = - \frac{y x}{(x + 2) (x - 2)} + \frac{18 x}{2 y (x + 2) (x - 2)}$

Step 3.5.5.3.1.4

Cancel the common factor of $18$ and $2$ .

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Step 3.5.5.3.1.4.1

Factor $2$ out of $18 x$ .

$y' = - \frac{y x}{(x + 2) (x - 2)} + \frac{2 (9 x)}{2 y (x + 2) (x - 2)}$

Step 3.5.5.3.1.4.2

Cancel the common factors.

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Step 3.5.5.3.1.4.2.1

Factor $2$ out of $2 y (x + 2) (x - 2)$ .

$y' = - \frac{y x}{(x + 2) (x - 2)} + \frac{2 (9 x)}{2 ((y (x + 2)) (x - 2))}$

Step 3.5.5.3.1.4.2.2

Cancel the common factor.

$y' = - \frac{y x}{(x + 2) (x - 2)} + \frac{2 (9 x)}{2 ((y (x + 2)) (x - 2))}$

Step 3.5.5.3.1.4.2.3

Rewrite the expression.

$y' = - \frac{y x}{(x + 2) (x - 2)} + \frac{9 x}{(y (x + 2)) (x - 2)}$

$y' = - \frac{y x}{(x + 2) (x - 2)} + \frac{9 x}{y (x + 2) (x - 2)}$

Step 3.6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{y x}{(x + 2) (x - 2)} + \frac{9 x}{y (x + 2) (x - 2)}$

Step 4

Set the derivative equal to $0$ then solve the equation $- \frac{y x}{(x + 2) (x - 2)} + \frac{9 x}{y (x + 2) (x - 2)} = 0$ .

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(x + 2) (x - 2), y (x + 2) (x - 2), 1$

Step 4.1.2

Since $(x + 2) (x - 2), y (x + 2) (x - 2), 1$ contains both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.

Steps to find the LCM for $(x + 2) (x - 2), y (x + 2) (x - 2), 1$ are:

1. Find the LCM for the numeric part $1, 1, 1$ .

2. Find the LCM for the variable part $y^{1}$ .

3. Find the LCM for the compound variable part $x + 2, x - 2, x + 2, x - 2$ .

4. Multiply each LCM together.

Step 4.1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 4.1.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 4.1.5

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 4.1.6

The factor for $y^{1}$ is $y$ itself.

$y^{1} = y$

$y$ occurs $1$ time.

Step 4.1.7

The LCM of $y^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y$

Step 4.1.8

The factor for $x + 2$ is $x + 2$ itself.

$(x + 2) = x + 2$

$(x + 2)$ occurs $1$ time.

Step 4.1.9

The factor for $x - 2$ is $x - 2$ itself.

$(x - 2) = x - 2$

$(x - 2)$ occurs $1$ time.

Step 4.1.10

The factor for $x + 2$ is $x + 2$ itself.

$(x + 2) = x + 2$

$(x + 2)$ occurs $1$ time.

Step 4.1.11

The factor for $x - 2$ is $x - 2$ itself.

$(x - 2) = x - 2$

$(x - 2)$ occurs $1$ time.

Step 4.1.12

The LCM of $x + 2, x - 2, x + 2, x - 2$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x + 2) (x - 2)$

Step 4.1.13

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$y (x + 2) (x - 2)$

Step 4.2

Multiply each term in $- \frac{y x}{(x + 2) (x - 2)} + \frac{9 x}{y (x + 2) (x - 2)} = 0$ by $y (x + 2) (x - 2)$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $- \frac{y x}{(x + 2) (x - 2)} + \frac{9 x}{y (x + 2) (x - 2)} = 0$ by $y (x + 2) (x - 2)$ .

$- \frac{y x}{(x + 2) (x - 2)} (y (x + 2) (x - 2)) + \frac{9 x}{y (x + 2) (x - 2)} (y (x + 2) (x - 2)) = 0 (y (x + 2) (x - 2))$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Cancel the common factor of $(x + 2) (x - 2)$ .

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Step 4.2.2.1.1.1

Move the leading negative in $- \frac{y x}{(x + 2) (x - 2)}$ into the numerator.

$\frac{- y x}{(x + 2) (x - 2)} (y (x + 2) (x - 2)) + \frac{9 x}{y (x + 2) (x - 2)} (y (x + 2) (x - 2)) = 0 (y (x + 2) (x - 2))$

Step 4.2.2.1.1.2

Factor $(x + 2) (x - 2)$ out of $y (x + 2) (x - 2)$ .

$\frac{- y x}{(x + 2) (x - 2)} ((x + 2) (x - 2) (y)) + \frac{9 x}{y (x + 2) (x - 2)} (y (x + 2) (x - 2)) = 0 (y (x + 2) (x - 2))$

Step 4.2.2.1.1.3

Cancel the common factor.

$\frac{- y x}{(x + 2) (x - 2)} ((x + 2) (x - 2) y) + \frac{9 x}{y (x + 2) (x - 2)} (y (x + 2) (x - 2)) = 0 (y (x + 2) (x - 2))$

Step 4.2.2.1.1.4

Rewrite the expression.

$- y x y + \frac{9 x}{y (x + 2) (x - 2)} (y (x + 2) (x - 2)) = 0 (y (x + 2) (x - 2))$

Step 4.2.2.1.2

Raise $y$ to the power of $1$ .

$- 1 x (y^{1} y) + \frac{9 x}{y (x + 2) (x - 2)} (y (x + 2) (x - 2)) = 0 (y (x + 2) (x - 2))$

Step 4.2.2.1.3

Raise $y$ to the power of $1$ .

$- 1 x (y^{1} y^{1}) + \frac{9 x}{y (x + 2) (x - 2)} (y (x + 2) (x - 2)) = 0 (y (x + 2) (x - 2))$

Step 4.2.2.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 1 x y^{1 + 1} + \frac{9 x}{y (x + 2) (x - 2)} (y (x + 2) (x - 2)) = 0 (y (x + 2) (x - 2))$

Step 4.2.2.1.5

Add $1$ and $1$ .

$- 1 x y^{2} + \frac{9 x}{y (x + 2) (x - 2)} (y (x + 2) (x - 2)) = 0 (y (x + 2) (x - 2))$

Step 4.2.2.1.6

Rewrite $- 1 x$ as $- x$ .

$- x y^{2} + \frac{9 x}{y (x + 2) (x - 2)} (y (x + 2) (x - 2)) = 0 (y (x + 2) (x - 2))$

Step 4.2.2.1.7

Cancel the common factor of $y (x + 2) (x - 2)$ .

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Step 4.2.2.1.7.1

Cancel the common factor.

$- x y^{2} + \frac{9 x}{y (x + 2) (x - 2)} (y (x + 2) (x - 2)) = 0 (y (x + 2) (x - 2))$

Step 4.2.2.1.7.2

Rewrite the expression.

$- x y^{2} + 9 x = 0 (y (x + 2) (x - 2))$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify by multiplying through.

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Step 4.2.3.1.1

Apply the distributive property.

$- x y^{2} + 9 x = 0 ((y x + y \cdot 2) (x - 2))$

Step 4.2.3.1.2

Move $2$ to the left of $y$ .

$- x y^{2} + 9 x = 0 ((y x + 2 \cdot y) (x - 2))$

Step 4.2.3.2

Expand $(y x + 2 y) (x - 2)$ using the FOIL Method.

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Step 4.2.3.2.1

Apply the distributive property.

$- x y^{2} + 9 x = 0 (y x (x - 2) + 2 y (x - 2))$

Step 4.2.3.2.2

Apply the distributive property.

$- x y^{2} + 9 x = 0 (y x \cdot x + y x \cdot - 2 + 2 y (x - 2))$

Step 4.2.3.2.3

Apply the distributive property.

$- x y^{2} + 9 x = 0 (y x \cdot x + y x \cdot - 2 + 2 y x + 2 y \cdot - 2)$

Step 4.2.3.3

Simplify terms.

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Step 4.2.3.3.1

Combine the opposite terms in $y x \cdot x + y x \cdot - 2 + 2 y x + 2 y \cdot - 2$ .

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Step 4.2.3.3.1.1

Reorder the factors in the terms $y x \cdot - 2$ and $2 y x$ .

$- x y^{2} + 9 x = 0 (y x \cdot x - 2 x y + 2 x y + 2 y \cdot - 2)$

Step 4.2.3.3.1.2

Add $- 2 x y$ and $2 x y$ .

$- x y^{2} + 9 x = 0 (y x \cdot x + 0 + 2 y \cdot - 2)$

Step 4.2.3.3.1.3

Add $y x \cdot x$ and $0$ .

$- x y^{2} + 9 x = 0 (y x \cdot x + 2 y \cdot - 2)$

Step 4.2.3.3.2

Simplify each term.

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Step 4.2.3.3.2.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.2.3.3.2.1.1

Move $x$ .

$- x y^{2} + 9 x = 0 (y (x \cdot x) + 2 y \cdot - 2)$

Step 4.2.3.3.2.1.2

Multiply $x$ by $x$ .

$- x y^{2} + 9 x = 0 (y x^{2} + 2 y \cdot - 2)$

Step 4.2.3.3.2.2

Multiply $- 2$ by $2$ .

$- x y^{2} + 9 x = 0 (y x^{2} - 4 y)$

Step 4.2.3.3.3

Multiply $0$ by $y x^{2} - 4 y$ .

$- x y^{2} + 9 x = 0$

Step 4.3

Solve the equation.

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Step 4.3.1

Factor $x$ out of $- x y^{2} + 9 x$ .

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Step 4.3.1.1

Factor $x$ out of $- x y^{2}$ .

$x (- 1 y^{2}) + 9 x = 0$

Step 4.3.1.2

Factor $x$ out of $9 x$ .

$x (- 1 y^{2}) + x \cdot 9 = 0$

Step 4.3.1.3

Factor $x$ out of $x (- 1 y^{2}) + x \cdot 9$ .

$x (- 1 y^{2} + 9) = 0$

Step 4.3.2

Rewrite $9$ as $3^{2}$ .

$x (- 1 y^{2} + 3^{2}) = 0$

Step 4.3.3

Reorder $- 1 y^{2}$ and $3^{2}$ .

$x (3^{2} - 1 y^{2}) = 0$

Step 4.3.4

Factor.

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Step 4.3.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = y$ .

$x ((3 + y) (3 - y)) = 0$

Step 4.3.4.2

Remove unnecessary parentheses.

$x (3 + y) (3 - y) = 0$

Step 4.3.5

Divide each term in $x (3 + y) (3 - y) = 0$ by $(3 + y) (3 - y)$ and simplify.

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Step 4.3.5.1

Divide each term in $x (3 + y) (3 - y) = 0$ by $(3 + y) (3 - y)$ .

$\frac{x (3 + y) (3 - y)}{(3 + y) (3 - y)} = \frac{0}{(3 + y) (3 - y)}$

Step 4.3.5.2

Simplify the left side.

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Step 4.3.5.2.1

Cancel the common factor of $3 + y$ .

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Step 4.3.5.2.1.1

Cancel the common factor.

$\frac{x (3 + y) (3 - y)}{(3 + y) (3 - y)} = \frac{0}{(3 + y) (3 - y)}$

Step 4.3.5.2.1.2

Rewrite the expression.

$\frac{x (3 - y)}{3 - y} = \frac{0}{(3 + y) (3 - y)}$

Step 4.3.5.2.2

Cancel the common factor of $3 - y$ .

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Step 4.3.5.2.2.1

Cancel the common factor.

$\frac{x (3 - y)}{3 - y} = \frac{0}{(3 + y) (3 - y)}$

Step 4.3.5.2.2.2

Divide $x$ by $1$ .

$x = \frac{0}{(3 + y) (3 - y)}$

Step 4.3.5.3

Simplify the right side.

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Step 4.3.5.3.1

Divide $0$ by $(3 + y) (3 - y)$ .

$x = 0$

Step 5

Solve the function $f (x) = \frac{3 x \sqrt{(x + 2) (x - 2)}}{(x + 2) (x - 2)}$ at $x = 0$ .

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Step 5.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{3 (0) \sqrt{((0) + 2) ((0) - 2)}}{((0) + 2) ((0) - 2)}$

Step 5.2

Simplify the result.

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Step 5.2.1

Reduce the expression by cancelling the common factors.

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Step 5.2.1.1

Cancel the common factor of $0$ and $(0) + 2$ .

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Step 5.2.1.1.1

Factor $2$ out of $3 (0) \sqrt{((0) + 2) ((0) - 2)}$ .

$f (0) = \frac{2 (3 \cdot ((0) \sqrt{((0) + 2) ((0) - 2)}))}{((0) + 2) ((0) - 2)}$

Step 5.2.1.1.2

Cancel the common factors.

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Step 5.2.1.1.2.1

Factor $2$ out of $((0) + 2) ((0) - 2)$ .

$f (0) = \frac{2 (3 \cdot ((0) \sqrt{((0) + 2) ((0) - 2)}))}{2 ((0 + 1) ((0) - 2))}$

Step 5.2.1.1.2.2

Cancel the common factor.

$f (0) = \frac{2 (3 \cdot ((0) \sqrt{((0) + 2) ((0) - 2)}))}{2 ((0 + 1) ((0) - 2))}$

Step 5.2.1.1.2.3

Rewrite the expression.

$f (0) = \frac{3 \cdot ((0) \sqrt{((0) + 2) ((0) - 2)})}{(0 + 1) ((0) - 2)}$

Step 5.2.1.2

Cancel the common factor of $0$ and $(0) - 2$ .

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Step 5.2.1.2.1

Factor $2$ out of $3 \cdot (0) \sqrt{((0) + 2) ((0) - 2)}$ .

$f (0) = \frac{2 (3 \cdot ((0) \sqrt{((0) + 2) ((0) - 2)}))}{(0 + 1) ((0) - 2)}$

Step 5.2.1.2.2

Cancel the common factors.

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Step 5.2.1.2.2.1

Factor $2$ out of $(0 + 1) ((0) - 2)$ .

$f (0) = \frac{2 (3 \cdot ((0) \sqrt{((0) + 2) ((0) - 2)}))}{2 ((0 + 1) (0 - 1))}$

Step 5.2.1.2.2.2

Cancel the common factor.

$f (0) = \frac{2 (3 \cdot ((0) \sqrt{((0) + 2) ((0) - 2)}))}{2 ((0 + 1) (0 - 1))}$

Step 5.2.1.2.2.3

Rewrite the expression.

$f (0) = \frac{3 \cdot ((0) \sqrt{((0) + 2) ((0) - 2)})}{(0 + 1) (0 - 1)}$

Step 5.2.2

Simplify the numerator.

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Step 5.2.2.1

Multiply $3$ by $0$ .

$f (0) = \frac{0 \sqrt{(0 + 2) (0 - 2)}}{(0 + 1) (0 - 1)}$

Step 5.2.2.2

Multiply $0$ by $\sqrt{(0 + 2) (0 - 2)}$ .

$f (0) = \frac{0}{(0 + 1) (0 - 1)}$

Step 5.2.3

Simplify the denominator.

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Step 5.2.3.1

Rewrite $0$ as $- 1 (0)$ .

$f (0) = \frac{0}{(- 1 \cdot 0 + 1) (0 - 1)}$

Step 5.2.3.2

Rewrite $1$ as $- 1 (- 1)$ .

$f (0) = \frac{0}{(- 1 \cdot 0 - 1 \cdot - 1) (0 - 1)}$

Step 5.2.3.3

Factor $- 1$ out of $- 1 \cdot 0 - 1 \cdot - 1$ .

$f (0) = \frac{0}{- 1 \cdot ((0 - 1) (0 - 1))}$

Step 5.2.3.4

Raise $0 - 1$ to the power of $1$ .

$f (0) = \frac{0}{- 1 \cdot ((0 - 1) (0 - 1))}$

Step 5.2.3.5

Raise $0 - 1$ to the power of $1$ .

$f (0) = \frac{0}{- 1 \cdot ((0 - 1) (0 - 1))}$

Step 5.2.3.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (0) = \frac{0}{- 1 \cdot {(0 - 1)}^{1 + 1}}$

Step 5.2.3.7

Add $1$ and $1$ .

$f (0) = \frac{0}{- 1 \cdot {(0 - 1)}^{2}}$

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

Subtract $1$ from $0$ .

$f (0) = \frac{0}{- 1 {(- 1)}^{2}}$

Step 5.2.4.2

Raise $- 1$ to the power of $2$ .

$f (0) = \frac{0}{- 1 \cdot 1}$

Step 5.2.5

Simplify the expression.

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Step 5.2.5.1

Multiply $- 1$ by $1$ .

$f (0) = \frac{0}{- 1}$

Step 5.2.5.2

Divide $0$ by $- 1$ .

$f (0) = 0$

Step 5.2.6

The final answer is $0$ .

$0$

Step 6

The horizontal tangent lines are $y = 0, y = 0$

$y = 0, y = 0$

Step 7