Calculus Examples

$f (x) = \frac{x^{3}}{x^{2} - 4}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{3}$ and $g (x) = x^{2} - 4$ .

$f' (x) = \frac{(x^{2} - 4) \frac{d}{d x} (x^{3}) - x^{3} \frac{d}{d x} (x^{2} - 4)}{{(x^{2} - 4)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f' (x) = \frac{(x^{2} - 4) (3 x^{2}) - x^{3} \frac{d}{d x} (x^{2} - 4)}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.2

Move $3$ to the left of $x^{2} - 4$ .

$f' (x) = \frac{3 \cdot ((x^{2} - 4) x^{2}) - x^{3} \frac{d}{d x} (x^{2} - 4)}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.3

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

$f' (x) = \frac{3 (x^{2} - 4) x^{2} - x^{3} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 4))}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.4

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

$f' (x) = \frac{3 (x^{2} - 4) x^{2} - x^{3} (2 x + \frac{d}{d x} (- 4))}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$f' (x) = \frac{3 (x^{2} - 4) x^{2} - x^{3} (2 x + 0)}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.6

Simplify the expression.

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Step 1.1.2.6.1

Add $2 x$ and $0$ .

$f' (x) = \frac{3 (x^{2} - 4) x^{2} - x^{3} (2 x)}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.6.2

Multiply $2$ by $- 1$ .

$f' (x) = \frac{3 (x^{2} - 4) x^{2} - 2 x^{3} x}{{(x^{2} - 4)}^{2}}$

Step 1.1.3

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

$f' (x) = \frac{3 (x^{2} - 4) x^{2} - 2 (x \cdot x^{3})}{{(x^{2} - 4)}^{2}}$

Step 1.1.4

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

$f' (x) = \frac{3 (x^{2} - 4) x^{2} - 2 x^{1 + 3}}{{(x^{2} - 4)}^{2}}$

Step 1.1.5

Add $1$ and $3$ .

$f' (x) = \frac{3 (x^{2} - 4) x^{2} - 2 x^{4}}{{(x^{2} - 4)}^{2}}$

Step 1.1.6

Simplify.

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Step 1.1.6.1

Apply the distributive property.

$f' (x) = \frac{(3 x^{2} + 3 \cdot - 4) x^{2} - 2 x^{4}}{{(x^{2} - 4)}^{2}}$

Step 1.1.6.2

Apply the distributive property.

$f' (x) = \frac{3 x^{2} x^{2} + 3 \cdot (- 4 x^{2}) - 2 x^{4}}{{(x^{2} - 4)}^{2}}$

Step 1.1.6.3

Simplify the numerator.

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Step 1.1.6.3.1

Simplify each term.

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Step 1.1.6.3.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.1.6.3.1.1.1

Move $x^{2}$ .

$f' (x) = \frac{3 (x^{2} x^{2}) + 3 \cdot (- 4 x^{2}) - 2 x^{4}}{{(x^{2} - 4)}^{2}}$

Step 1.1.6.3.1.1.2

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

$f' (x) = \frac{3 x^{2 + 2} + 3 \cdot (- 4 x^{2}) - 2 x^{4}}{{(x^{2} - 4)}^{2}}$

Step 1.1.6.3.1.1.3

Add $2$ and $2$ .

$f' (x) = \frac{3 x^{4} + 3 \cdot (- 4 x^{2}) - 2 x^{4}}{{(x^{2} - 4)}^{2}}$

Step 1.1.6.3.1.2

Multiply $3$ by $- 4$ .

$f' (x) = \frac{3 x^{4} - 12 x^{2} - 2 x^{4}}{{(x^{2} - 4)}^{2}}$

Step 1.1.6.3.2

Subtract $2 x^{4}$ from $3 x^{4}$ .

$f' (x) = \frac{x^{4} - 12 x^{2}}{{(x^{2} - 4)}^{2}}$

Step 1.1.6.4

Factor $x^{2}$ out of $x^{4} - 12 x^{2}$ .

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Step 1.1.6.4.1

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

$f' (x) = \frac{x^{2} x^{2} - 12 x^{2}}{{(x^{2} - 4)}^{2}}$

Step 1.1.6.4.2

Factor $x^{2}$ out of $- 12 x^{2}$ .

$f' (x) = \frac{x^{2} x^{2} + x^{2} \cdot - 12}{{(x^{2} - 4)}^{2}}$

Step 1.1.6.4.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 12$ .

$f' (x) = \frac{x^{2} (x^{2} - 12)}{{(x^{2} - 4)}^{2}}$

Step 1.1.6.5

Simplify the denominator.

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Step 1.1.6.5.1

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

$f' (x) = \frac{x^{2} (x^{2} - 12)}{{(x^{2} - 2^{2})}^{2}}$

Step 1.1.6.5.2

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 $b = 2$ .

$f' (x) = \frac{x^{2} (x^{2} - 12)}{{((x + 2) (x - 2))}^{2}}$

Step 1.1.6.5.3

Apply the product rule to $(x + 2) (x - 2)$ .

$f' (x) = \frac{x^{2} (x^{2} - 12)}{{(x + 2)}^{2} {(x - 2)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x^{2} (x^{2} - 12)}{{(x + 2)}^{2} {(x - 2)}^{2}}$ .

$\frac{x^{2} (x^{2} - 12)}{{(x + 2)}^{2} {(x - 2)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{x^{2} (x^{2} - 12)}{{(x + 2)}^{2} {(x - 2)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{x^{2} (x^{2} - 12)}{{(x + 2)}^{2} {(x - 2)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$x^{2} (x^{2} - 12) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$x^{2} - 12 = 0$

Step 2.3.2

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 2.3.2.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.3.2.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.3.2.2.1

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

$x = \pm \sqrt{0}$

Step 2.3.2.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.3.2.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.3.2.2.2.2

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

$x = \pm 0$

Step 2.3.2.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.3.3

Set $x^{2} - 12$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $x^{2} - 12$ equal to $0$ .

$x^{2} - 12 = 0$

Step 2.3.3.2

Solve $x^{2} - 12 = 0$ for $x$ .

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Step 2.3.3.2.1

Add $12$ to both sides of the equation.

$x^{2} = 12$

Step 2.3.3.2.2

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

$x = \pm \sqrt{12}$

Step 2.3.3.2.3

Simplify $\pm \sqrt{12}$ .

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Step 2.3.3.2.3.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.3.3.2.3.1.1

Factor $4$ out of $12$ .

$x = \pm \sqrt{4 (3)}$

Step 2.3.3.2.3.1.2

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

$x = \pm \sqrt{2^{2} \cdot 3}$

Step 2.3.3.2.3.2

Pull terms out from under the radical.

$x = \pm 2 \sqrt{3}$

Step 2.3.3.2.4

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

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Step 2.3.3.2.4.1

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

$x = 2 \sqrt{3}$

Step 2.3.3.2.4.2

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

$x = - 2 \sqrt{3}$

Step 2.3.3.2.4.3

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

$x = 2 \sqrt{3}, - 2 \sqrt{3}$

Step 2.3.4

The final solution is all the values that make $x^{2} (x^{2} - 12) = 0$ true.

$x = 0, 2 \sqrt{3}, - 2 \sqrt{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{x^{2} (x^{2} - 12)}{{(x + 2)}^{2} {(x - 2)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 3.2

Solve for $x$ .

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Step 3.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x + 2)}^{2} = 0$

${(x - 2)}^{2} = 0$

Step 3.2.2

Set ${(x + 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.2.1

Set ${(x + 2)}^{2}$ equal to $0$ .

${(x + 2)}^{2} = 0$

Step 3.2.2.2

Solve ${(x + 2)}^{2} = 0$ for $x$ .

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Step 3.2.2.2.1

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.2.2.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.2.3

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.3.1

Set ${(x - 2)}^{2}$ equal to $0$ .

${(x - 2)}^{2} = 0$

Step 3.2.3.2

Solve ${(x - 2)}^{2} = 0$ for $x$ .

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Step 3.2.3.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.2.3.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.2.4

The final solution is all the values that make ${(x + 2)}^{2} {(x - 2)}^{2} = 0$ true.

$x = - 2, 2$

Step 3.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 2, x = 2$

Step 4

Evaluate $\frac{x^{3}}{x^{2} - 4}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$\frac{{(0)}^{3}}{{(0)}^{2} - 4}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0^{2} - 4}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0 - 4}$

Step 4.1.2.2.2

Subtract $4$ from $0$ .

$\frac{0}{- 4}$

Step 4.1.2.3

Divide $0$ by $- 4$ .

$0$

Step 4.2

Evaluate at $x = 2 \sqrt{3}$ .

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Step 4.2.1

Substitute $2 \sqrt{3}$ for $x$ .

$\frac{{(2 \sqrt{3})}^{3}}{{(2 \sqrt{3})}^{2} - 4}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

Apply the product rule to $2 \sqrt{3}$ .

$\frac{2^{3} {\sqrt{3}}^{3}}{{(2 \sqrt{3})}^{2} - 4}$

Step 4.2.2.1.2

Raise $2$ to the power of $3$ .

$\frac{8 {\sqrt{3}}^{3}}{{(2 \sqrt{3})}^{2} - 4}$

Step 4.2.2.1.3

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$\frac{8 \sqrt{3^{3}}}{{(2 \sqrt{3})}^{2} - 4}$

Step 4.2.2.1.4

Raise $3$ to the power of $3$ .

$\frac{8 \sqrt{27}}{{(2 \sqrt{3})}^{2} - 4}$

Step 4.2.2.1.5

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 4.2.2.1.5.1

Factor $9$ out of $27$ .

$\frac{8 \sqrt{9 (3)}}{{(2 \sqrt{3})}^{2} - 4}$

Step 4.2.2.1.5.2

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

$\frac{8 \sqrt{3^{2} \cdot 3}}{{(2 \sqrt{3})}^{2} - 4}$

Step 4.2.2.1.6

Pull terms out from under the radical.

$\frac{8 \cdot 3 \sqrt{3}}{{(2 \sqrt{3})}^{2} - 4}$

Step 4.2.2.1.7

Multiply $8$ by $3$ .

$\frac{24 \sqrt{3}}{{(2 \sqrt{3})}^{2} - 4}$

Step 4.2.2.2

Simplify the denominator.

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Step 4.2.2.2.1

Apply the product rule to $2 \sqrt{3}$ .

$\frac{24 \sqrt{3}}{2^{2} {\sqrt{3}}^{2} - 4}$

Step 4.2.2.2.2

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

$\frac{24 \sqrt{3}}{4 {\sqrt{3}}^{2} - 4}$

Step 4.2.2.2.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.2.2.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{24 \sqrt{3}}{4 {(3^{\frac{1}{2}})}^{2} - 4}$

Step 4.2.2.2.3.2

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

$\frac{24 \sqrt{3}}{4 \cdot 3^{\frac{1}{2} \cdot 2} - 4}$

Step 4.2.2.2.3.3

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

$\frac{24 \sqrt{3}}{4 \cdot 3^{\frac{2}{2}} - 4}$

Step 4.2.2.2.3.4

Cancel the common factor of $2$ .

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Step 4.2.2.2.3.4.1

Cancel the common factor.

$\frac{24 \sqrt{3}}{4 \cdot 3^{\frac{2}{2}} - 4}$

Step 4.2.2.2.3.4.2

Rewrite the expression.

$\frac{24 \sqrt{3}}{4 \cdot 3^{1} - 4}$

Step 4.2.2.2.3.5

Evaluate the exponent.

$\frac{24 \sqrt{3}}{4 \cdot 3 - 4}$

Step 4.2.2.2.4

Multiply $4$ by $3$ .

$\frac{24 \sqrt{3}}{12 - 4}$

Step 4.2.2.2.5

Subtract $4$ from $12$ .

$\frac{24 \sqrt{3}}{8}$

Step 4.2.2.3

Cancel the common factor of $24$ and $8$ .

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Step 4.2.2.3.1

Factor $8$ out of $24 \sqrt{3}$ .

$\frac{8 (3 \sqrt{3})}{8}$

Step 4.2.2.3.2

Cancel the common factors.

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Step 4.2.2.3.2.1

Factor $8$ out of $8$ .

$\frac{8 (3 \sqrt{3})}{8 (1)}$

Step 4.2.2.3.2.2

Cancel the common factor.

$\frac{8 (3 \sqrt{3})}{8 \cdot 1}$

Step 4.2.2.3.2.3

Rewrite the expression.

$\frac{3 \sqrt{3}}{1}$

Step 4.2.2.3.2.4

Divide $3 \sqrt{3}$ by $1$ .

$3 \sqrt{3}$

Step 4.3

Evaluate at $x = - 2 \sqrt{3}$ .

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Step 4.3.1

Substitute $- 2 \sqrt{3}$ for $x$ .

$\frac{{(- 2 \sqrt{3})}^{3}}{{(- 2 \sqrt{3})}^{2} - 4}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify the numerator.

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Step 4.3.2.1.1

Apply the product rule to $- 2 \sqrt{3}$ .

$\frac{{(- 2)}^{3} {\sqrt{3}}^{3}}{{(- 2 \sqrt{3})}^{2} - 4}$

Step 4.3.2.1.2

Raise $- 2$ to the power of $3$ .

$\frac{- 8 {\sqrt{3}}^{3}}{{(- 2 \sqrt{3})}^{2} - 4}$

Step 4.3.2.1.3

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$\frac{- 8 \sqrt{3^{3}}}{{(- 2 \sqrt{3})}^{2} - 4}$

Step 4.3.2.1.4

Raise $3$ to the power of $3$ .

$\frac{- 8 \sqrt{27}}{{(- 2 \sqrt{3})}^{2} - 4}$

Step 4.3.2.1.5

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 4.3.2.1.5.1

Factor $9$ out of $27$ .

$\frac{- 8 \sqrt{9 (3)}}{{(- 2 \sqrt{3})}^{2} - 4}$

Step 4.3.2.1.5.2

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

$\frac{- 8 \sqrt{3^{2} \cdot 3}}{{(- 2 \sqrt{3})}^{2} - 4}$

Step 4.3.2.1.6

Pull terms out from under the radical.

$\frac{- 8 \cdot 3 \sqrt{3}}{{(- 2 \sqrt{3})}^{2} - 4}$

Step 4.3.2.1.7

Multiply $- 8$ by $3$ .

$\frac{- 24 \sqrt{3}}{{(- 2 \sqrt{3})}^{2} - 4}$

Step 4.3.2.2

Simplify the denominator.

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Step 4.3.2.2.1

Apply the product rule to $- 2 \sqrt{3}$ .

$\frac{- 24 \sqrt{3}}{{(- 2)}^{2} {\sqrt{3}}^{2} - 4}$

Step 4.3.2.2.2

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

$\frac{- 24 \sqrt{3}}{4 {\sqrt{3}}^{2} - 4}$

Step 4.3.2.2.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.3.2.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{- 24 \sqrt{3}}{4 {(3^{\frac{1}{2}})}^{2} - 4}$

Step 4.3.2.2.3.2

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

$\frac{- 24 \sqrt{3}}{4 \cdot 3^{\frac{1}{2} \cdot 2} - 4}$

Step 4.3.2.2.3.3

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

$\frac{- 24 \sqrt{3}}{4 \cdot 3^{\frac{2}{2}} - 4}$

Step 4.3.2.2.3.4

Cancel the common factor of $2$ .

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Step 4.3.2.2.3.4.1

Cancel the common factor.

$\frac{- 24 \sqrt{3}}{4 \cdot 3^{\frac{2}{2}} - 4}$

Step 4.3.2.2.3.4.2

Rewrite the expression.

$\frac{- 24 \sqrt{3}}{4 \cdot 3^{1} - 4}$

Step 4.3.2.2.3.5

Evaluate the exponent.

$\frac{- 24 \sqrt{3}}{4 \cdot 3 - 4}$

Step 4.3.2.2.4

Multiply $4$ by $3$ .

$\frac{- 24 \sqrt{3}}{12 - 4}$

Step 4.3.2.2.5

Subtract $4$ from $12$ .

$\frac{- 24 \sqrt{3}}{8}$

Step 4.3.2.3

Cancel the common factor of $- 24$ and $8$ .

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Step 4.3.2.3.1

Factor $8$ out of $- 24 \sqrt{3}$ .

$\frac{8 (- 3 \sqrt{3})}{8}$

Step 4.3.2.3.2

Cancel the common factors.

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Step 4.3.2.3.2.1

Factor $8$ out of $8$ .

$\frac{8 (- 3 \sqrt{3})}{8 (1)}$

Step 4.3.2.3.2.2

Cancel the common factor.

$\frac{8 (- 3 \sqrt{3})}{8 \cdot 1}$

Step 4.3.2.3.2.3

Rewrite the expression.

$\frac{- 3 \sqrt{3}}{1}$

Step 4.3.2.3.2.4

Divide $- 3 \sqrt{3}$ by $1$ .

$- 3 \sqrt{3}$

Step 4.4

Evaluate at $x = - 2$ .

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Step 4.4.1

Substitute $- 2$ for $x$ .

$\frac{{(- 2)}^{3}}{{(- 2)}^{2} - 4}$

Step 4.4.2

Simplify.

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Step 4.4.2.1

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

$\frac{{(- 2)}^{3}}{4 - 4}$

Step 4.4.2.2

Subtract $4$ from $4$ .

$\frac{{(- 2)}^{3}}{0}$

Step 4.4.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.5

Evaluate at $x = 2$ .

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Step 4.5.1

Substitute $2$ for $x$ .

$\frac{{(2)}^{3}}{{(2)}^{2} - 4}$

Step 4.5.2

Simplify.

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Step 4.5.2.1

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

$\frac{{(2)}^{3}}{4 - 4}$

Step 4.5.2.2

Subtract $4$ from $4$ .

$\frac{{(2)}^{3}}{0}$

Step 4.5.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.6

List all of the points.

$(0, 0), (2 \sqrt{3}, 3 \sqrt{3}), (- 2 \sqrt{3}, - 3 \sqrt{3})$

Step 5