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

The values which make the derivative equal to $0$ are $0, 2 \sqrt{3}, - 2 \sqrt{3}$ .

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

Step 4

Find where the derivative is undefined.

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Step 4.1

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 4.2

Solve for $x$ .

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Step 4.2.1

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 4.2.2

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

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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Step 4.2.2.2.1

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

$x + 2 = 0$

Step 4.2.2.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4.2.3

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

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Step 4.2.3.1

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

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

Step 4.2.3.2

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

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Step 4.2.3.2.1

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

$x - 2 = 0$

Step 4.2.3.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4.2.4

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

$x = - 2, 2$

Step 4.3

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 5

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

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

Step 6

Substitute a value from the interval $(- \infty, - 2 \sqrt{3})$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 4.4641016$ in the expression.

$f' (- 4.4641016) = \frac{{(- 4.4641016)}^{2} ({(- 4.4641016)}^{2} - 12)}{{((- 4.4641016) + 2)}^{2} {((- 4.4641016) - 2)}^{2}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Raise $- 4.4641016$ to the power of $2$ .

$f' (- 4.4641016) = \frac{{(- 4.4641016)}^{2} (19.92820309 - 12)}{{(- 4.4641016 + 2)}^{2} {(- 4.4641016 - 2)}^{2}}$

Step 6.2.1.2

Subtract $12$ from $19.92820309$ .

$f' (- 4.4641016) = \frac{{(- 4.4641016)}^{2} \cdot 7.92820309}{{(- 4.4641016 + 2)}^{2} {(- 4.4641016 - 2)}^{2}}$

Step 6.2.1.3

Raise $- 4.4641016$ to the power of $2$ .

$f' (- 4.4641016) = \frac{19.92820309 \cdot 7.92820309}{{(- 4.4641016 + 2)}^{2} {(- 4.4641016 - 2)}^{2}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Add $- 4.4641016$ and $2$ .

$f' (- 4.4641016) = \frac{19.92820309 \cdot 7.92820309}{{(- 2.4641016)}^{2} {(- 4.4641016 - 2)}^{2}}$

Step 6.2.2.2

Subtract $2$ from $- 4.4641016$ .

$f' (- 4.4641016) = \frac{19.92820309 \cdot 7.92820309}{{(- 2.4641016)}^{2} {(- 6.4641016)}^{2}}$

Step 6.2.2.3

Raise $- 2.4641016$ to the power of $2$ .

$f' (- 4.4641016) = \frac{19.92820309 \cdot 7.92820309}{6.07179669 {(- 6.4641016)}^{2}}$

Step 6.2.2.4

Raise $- 6.4641016$ to the power of $2$ .

$f' (- 4.4641016) = \frac{19.92820309 \cdot 7.92820309}{6.07179669 \cdot 41.78460949}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Multiply $19.92820309$ by $7.92820309$ .

$f' (- 4.4641016) = \frac{157.99484145}{6.07179669 \cdot 41.78460949}$

Step 6.2.3.2

Multiply $6.07179669$ by $41.78460949$ .

$f' (- 4.4641016) = \frac{157.99484145}{253.70765383}$

Step 6.2.3.3

Divide $157.99484145$ by $253.70765383$ .

$f' (- 4.4641016) = 0.62274369$

Step 6.2.4

The final answer is $0.62274369$ .

$0.62274369$

Step 6.3

At $x = - 4.4641016$ the derivative is $0.62274369$ . Since this is positive, the function is increasing on $(- \infty, - 2 \sqrt{3})$ .

Increasing on $(- \infty, - 2 \sqrt{3})$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(- 3.4641016, - 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 2.7320508$ in the expression.

$f' (- 2.7320508) = \frac{{(- 2.7320508)}^{2} ({(- 2.7320508)}^{2} - 12)}{{((- 2.7320508) + 2)}^{2} {((- 2.7320508) - 2)}^{2}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Raise $- 2.7320508$ to the power of $2$ .

$f' (- 2.7320508) = \frac{{(- 2.7320508)}^{2} (7.46410157 - 12)}{{(- 2.7320508 + 2)}^{2} {(- 2.7320508 - 2)}^{2}}$

Step 7.2.1.2

Subtract $12$ from $7.46410157$ .

$f' (- 2.7320508) = \frac{{(- 2.7320508)}^{2} \cdot - 4.53589842}{{(- 2.7320508 + 2)}^{2} {(- 2.7320508 - 2)}^{2}}$

Step 7.2.1.3

Raise $- 2.7320508$ to the power of $2$ .

$f' (- 2.7320508) = \frac{7.46410157 \cdot - 4.53589842}{{(- 2.7320508 + 2)}^{2} {(- 2.7320508 - 2)}^{2}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Add $- 2.7320508$ and $2$ .

$f' (- 2.7320508) = \frac{7.46410157 \cdot - 4.53589842}{{(- 0.7320508)}^{2} {(- 2.7320508 - 2)}^{2}}$

Step 7.2.2.2

Subtract $2$ from $- 2.7320508$ .

$f' (- 2.7320508) = \frac{7.46410157 \cdot - 4.53589842}{{(- 0.7320508)}^{2} {(- 4.7320508)}^{2}}$

Step 7.2.2.3

Raise $- 0.7320508$ to the power of $2$ .

$f' (- 2.7320508) = \frac{7.46410157 \cdot - 4.53589842}{0.53589837 {(- 4.7320508)}^{2}}$

Step 7.2.2.4

Raise $- 4.7320508$ to the power of $2$ .

$f' (- 2.7320508) = \frac{7.46410157 \cdot - 4.53589842}{0.53589837 \cdot 22.39230477}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Multiply $7.46410157$ by $- 4.53589842$ .

$f' (- 2.7320508) = \frac{- 33.85640658}{0.53589837 \cdot 22.39230477}$

Step 7.2.3.2

Multiply $0.53589837$ by $22.39230477$ .

$f' (- 2.7320508) = \frac{- 33.85640658}{11.99999971}$

Step 7.2.3.3

Divide $- 33.85640658$ by $11.99999971$ .

$f' (- 2.7320508) = - 2.82136728$

Step 7.2.4

The final answer is $- 2.82136728$ .

$- 2.82136728$

Step 7.3

At $x = - 2.7320508$ the derivative is $- 2.82136728$ . Since this is negative, the function is decreasing on $(- 3.4641016, - 2)$ .

Decreasing on $(- 2 \sqrt{3}, - 2)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(- 2, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = \frac{{(- 1)}^{2} ({(- 1)}^{2} - 12)}{{((- 1) + 2)}^{2} {((- 1) - 2)}^{2}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify the numerator.

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Step 8.2.1.1

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

$f' (- 1) = \frac{{(- 1)}^{2} (1 - 12)}{{(- 1 + 2)}^{2} {(- 1 - 2)}^{2}}$

Step 8.2.1.2

Subtract $12$ from $1$ .

$f' (- 1) = \frac{{(- 1)}^{2} \cdot - 11}{{(- 1 + 2)}^{2} {(- 1 - 2)}^{2}}$

Step 8.2.1.3

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

$f' (- 1) = \frac{1 \cdot - 11}{{(- 1 + 2)}^{2} {(- 1 - 2)}^{2}}$

Step 8.2.1.4

Multiply $- 11$ by $1$ .

$f' (- 1) = \frac{- 11}{{(- 1 + 2)}^{2} {(- 1 - 2)}^{2}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Add $- 1$ and $2$ .

$f' (- 1) = \frac{- 11}{1^{2} {(- 1 - 2)}^{2}}$

Step 8.2.2.2

Subtract $2$ from $- 1$ .

$f' (- 1) = \frac{- 11}{1^{2} {(- 3)}^{2}}$

Step 8.2.2.3

One to any power is one.

$f' (- 1) = \frac{- 11}{1 {(- 3)}^{2}}$

Step 8.2.2.4

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

$f' (- 1) = \frac{- 11}{1 \cdot 9}$

Step 8.2.2.5

Multiply $9$ by $1$ .

$f' (- 1) = \frac{- 11}{9}$

Step 8.2.3

Move the negative in front of the fraction.

$f' (- 1) = - \frac{11}{9}$

Step 8.2.4

The final answer is $- \frac{11}{9}$ .

$- \frac{11}{9}$

Step 8.3

At $x = - 1$ the derivative is $- \frac{11}{9}$ . Since this is negative, the function is decreasing on $(- 2, 0)$ .

Decreasing on $(- 2, 0)$ since $f' (x) < 0$

Step 9

Substitute a value from the interval $(0, 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = \frac{{(1)}^{2} ({(1)}^{2} - 12)}{{((1) + 2)}^{2} {((1) - 2)}^{2}}$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify the numerator.

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Step 9.2.1.1

One to any power is one.

$f' (1) = \frac{1^{2} (1 - 12)}{{(1 + 2)}^{2} {(1 - 2)}^{2}}$

Step 9.2.1.2

Subtract $12$ from $1$ .

$f' (1) = \frac{1^{2} \cdot - 11}{{(1 + 2)}^{2} {(1 - 2)}^{2}}$

Step 9.2.1.3

One to any power is one.

$f' (1) = \frac{1 \cdot - 11}{{(1 + 2)}^{2} {(1 - 2)}^{2}}$

Step 9.2.1.4

Multiply $- 11$ by $1$ .

$f' (1) = \frac{- 11}{{(1 + 2)}^{2} {(1 - 2)}^{2}}$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

Add $1$ and $2$ .

$f' (1) = \frac{- 11}{3^{2} {(1 - 2)}^{2}}$

Step 9.2.2.2

Subtract $2$ from $1$ .

$f' (1) = \frac{- 11}{3^{2} {(- 1)}^{2}}$

Step 9.2.2.3

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

$f' (1) = \frac{- 11}{9 {(- 1)}^{2}}$

Step 9.2.2.4

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

$f' (1) = \frac{- 11}{9 \cdot 1}$

Step 9.2.3

Simplify the expression.

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Step 9.2.3.1

Multiply $9$ by $1$ .

$f' (1) = \frac{- 11}{9}$

Step 9.2.3.2

Move the negative in front of the fraction.

$f' (1) = - \frac{11}{9}$

Step 9.2.4

The final answer is $- \frac{11}{9}$ .

$- \frac{11}{9}$

Step 9.3

At $x = 1$ the derivative is $- \frac{11}{9}$ . Since this is negative, the function is decreasing on $(0, 2)$ .

Decreasing on $(0, 2)$ since $f' (x) < 0$

Step 10

Substitute a value from the interval $(2, 2 \sqrt{3})$ into the derivative to determine if the function is increasing or decreasing.

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Step 10.1

Replace the variable $x$ with $2.7320508$ in the expression.

$f' (2.7320508) = \frac{{(2.7320508)}^{2} ({(2.7320508)}^{2} - 12)}{{((2.7320508) + 2)}^{2} {((2.7320508) - 2)}^{2}}$

Step 10.2

Simplify the result.

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Step 10.2.1

Simplify the numerator.

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Step 10.2.1.1

Raise $2.7320508$ to the power of $2$ .

$f' (2.7320508) = \frac{{2.7320508}^{2} (7.46410157 - 12)}{{(2.7320508 + 2)}^{2} {(2.7320508 - 2)}^{2}}$

Step 10.2.1.2

Subtract $12$ from $7.46410157$ .

$f' (2.7320508) = \frac{{2.7320508}^{2} \cdot - 4.53589842}{{(2.7320508 + 2)}^{2} {(2.7320508 - 2)}^{2}}$

Step 10.2.1.3

Raise $2.7320508$ to the power of $2$ .

$f' (2.7320508) = \frac{7.46410157 \cdot - 4.53589842}{{(2.7320508 + 2)}^{2} {(2.7320508 - 2)}^{2}}$

Step 10.2.2

Simplify the denominator.

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Step 10.2.2.1

Add $2.7320508$ and $2$ .

$f' (2.7320508) = \frac{7.46410157 \cdot - 4.53589842}{{4.7320508}^{2} {(2.7320508 - 2)}^{2}}$

Step 10.2.2.2

Subtract $2$ from $2.7320508$ .

$f' (2.7320508) = \frac{7.46410157 \cdot - 4.53589842}{{4.7320508}^{2} \cdot {0.7320508}^{2}}$

Step 10.2.2.3

Raise $4.7320508$ to the power of $2$ .

$f' (2.7320508) = \frac{7.46410157 \cdot - 4.53589842}{22.39230477 \cdot {0.7320508}^{2}}$

Step 10.2.2.4

Raise $0.7320508$ to the power of $2$ .

$f' (2.7320508) = \frac{7.46410157 \cdot - 4.53589842}{22.39230477 \cdot 0.53589837}$

Step 10.2.3

Simplify the expression.

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Step 10.2.3.1

Multiply $7.46410157$ by $- 4.53589842$ .

$f' (2.7320508) = \frac{- 33.85640658}{22.39230477 \cdot 0.53589837}$

Step 10.2.3.2

Multiply $22.39230477$ by $0.53589837$ .

$f' (2.7320508) = \frac{- 33.85640658}{11.99999971}$

Step 10.2.3.3

Divide $- 33.85640658$ by $11.99999971$ .

$f' (2.7320508) = - 2.82136728$

Step 10.2.4

The final answer is $- 2.82136728$ .

$- 2.82136728$

Step 10.3

At $x = 2.7320508$ the derivative is $- 2.82136728$ . Since this is negative, the function is decreasing on $(2, 2 \sqrt{3})$ .

Decreasing on $(2, 2 \sqrt{3})$ since $f' (x) < 0$

Step 11

Substitute a value from the interval $(2 \sqrt{3}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 11.1

Replace the variable $x$ with $4.4641016$ in the expression.

$f' (4.4641016) = \frac{{(4.4641016)}^{2} ({(4.4641016)}^{2} - 12)}{{((4.4641016) + 2)}^{2} {((4.4641016) - 2)}^{2}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify the numerator.

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Step 11.2.1.1

Raise $4.4641016$ to the power of $2$ .

$f' (4.4641016) = \frac{{4.4641016}^{2} (19.92820309 - 12)}{{(4.4641016 + 2)}^{2} {(4.4641016 - 2)}^{2}}$

Step 11.2.1.2

Subtract $12$ from $19.92820309$ .

$f' (4.4641016) = \frac{{4.4641016}^{2} \cdot 7.92820309}{{(4.4641016 + 2)}^{2} {(4.4641016 - 2)}^{2}}$

Step 11.2.1.3

Raise $4.4641016$ to the power of $2$ .

$f' (4.4641016) = \frac{19.92820309 \cdot 7.92820309}{{(4.4641016 + 2)}^{2} {(4.4641016 - 2)}^{2}}$

Step 11.2.2

Simplify the denominator.

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Step 11.2.2.1

Add $4.4641016$ and $2$ .

$f' (4.4641016) = \frac{19.92820309 \cdot 7.92820309}{{6.4641016}^{2} {(4.4641016 - 2)}^{2}}$

Step 11.2.2.2

Subtract $2$ from $4.4641016$ .

$f' (4.4641016) = \frac{19.92820309 \cdot 7.92820309}{{6.4641016}^{2} \cdot {2.4641016}^{2}}$

Step 11.2.2.3

Raise $6.4641016$ to the power of $2$ .

$f' (4.4641016) = \frac{19.92820309 \cdot 7.92820309}{41.78460949 \cdot {2.4641016}^{2}}$

Step 11.2.2.4

Raise $2.4641016$ to the power of $2$ .

$f' (4.4641016) = \frac{19.92820309 \cdot 7.92820309}{41.78460949 \cdot 6.07179669}$

Step 11.2.3

Simplify the expression.

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Step 11.2.3.1

Multiply $19.92820309$ by $7.92820309$ .

$f' (4.4641016) = \frac{157.99484145}{41.78460949 \cdot 6.07179669}$

Step 11.2.3.2

Multiply $41.78460949$ by $6.07179669$ .

$f' (4.4641016) = \frac{157.99484145}{253.70765383}$

Step 11.2.3.3

Divide $157.99484145$ by $253.70765383$ .

$f' (4.4641016) = 0.62274369$

Step 11.2.4

The final answer is $0.62274369$ .

$0.62274369$

Step 11.3

At $x = 4.4641016$ the derivative is $0.62274369$ . Since this is positive, the function is increasing on $(2 \sqrt{3}, \infty)$ .

Increasing on $(2 \sqrt{3}, \infty)$ since $f' (x) > 0$

Step 12

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, - 2 \sqrt{3}), (2 \sqrt{3}, \infty)$

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

Step 13