Calculus Examples

$f (x) = \frac{x^{3}}{x^{2} - 9}$

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 $g (x) = x^{2} - 9$ .

$f' (x) = \frac{(x^{2} - 9) \frac{d}{d x} (x^{3}) - x^{3} \frac{d}{d x} (x^{2} - 9)}{{(x^{2} - 9)}^{2}}$

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 $n = 3$ .

$f' (x) = \frac{(x^{2} - 9) (3 x^{2}) - x^{3} \frac{d}{d x} (x^{2} - 9)}{{(x^{2} - 9)}^{2}}$

Step 1.1.2.2

Move $3$ to the left of $x^{2} - 9$ .

$f' (x) = \frac{3 \cdot ((x^{2} - 9) x^{2}) - x^{3} \frac{d}{d x} (x^{2} - 9)}{{(x^{2} - 9)}^{2}}$

Step 1.1.2.3

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

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - x^{3} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 9))}{{(x^{2} - 9)}^{2}}$

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 $n = 2$ .

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - x^{3} (2 x + \frac{d}{d x} (- 9))}{{(x^{2} - 9)}^{2}}$

Step 1.1.2.5

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - x^{3} (2 x + 0)}{{(x^{2} - 9)}^{2}}$

Step 1.1.2.6

Simplify the expression.

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Step 1.1.2.6.1

Add $2 x$ and $0$ .

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - x^{3} (2 x)}{{(x^{2} - 9)}^{2}}$

Step 1.1.2.6.2

Multiply $2$ by $- 1$ .

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - 2 x^{3} x}{{(x^{2} - 9)}^{2}}$

Step 1.1.3

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

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - 2 (x \cdot x^{3})}{{(x^{2} - 9)}^{2}}$

Step 1.1.4

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

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - 2 x^{1 + 3}}{{(x^{2} - 9)}^{2}}$

Step 1.1.5

Add $1$ and $3$ .

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - 2 x^{4}}{{(x^{2} - 9)}^{2}}$

Step 1.1.6

Simplify.

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Step 1.1.6.1

Apply the distributive property.

$f' (x) = \frac{(3 x^{2} + 3 \cdot - 9) x^{2} - 2 x^{4}}{{(x^{2} - 9)}^{2}}$

Step 1.1.6.2

Apply the distributive property.

$f' (x) = \frac{3 x^{2} x^{2} + 3 \cdot (- 9 x^{2}) - 2 x^{4}}{{(x^{2} - 9)}^{2}}$

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 $x^{2}$ .

$f' (x) = \frac{3 (x^{2} x^{2}) + 3 \cdot (- 9 x^{2}) - 2 x^{4}}{{(x^{2} - 9)}^{2}}$

Step 1.1.6.3.1.1.2

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

$f' (x) = \frac{3 x^{2 + 2} + 3 \cdot (- 9 x^{2}) - 2 x^{4}}{{(x^{2} - 9)}^{2}}$

Step 1.1.6.3.1.1.3

Add $2$ and $2$ .

$f' (x) = \frac{3 x^{4} + 3 \cdot (- 9 x^{2}) - 2 x^{4}}{{(x^{2} - 9)}^{2}}$

Step 1.1.6.3.1.2

Multiply $3$ by $- 9$ .

$f' (x) = \frac{3 x^{4} - 27 x^{2} - 2 x^{4}}{{(x^{2} - 9)}^{2}}$

Step 1.1.6.3.2

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

$f' (x) = \frac{x^{4} - 27 x^{2}}{{(x^{2} - 9)}^{2}}$

Step 1.1.6.4

Factor $x^{2}$ out of $x^{4} - 27 x^{2}$ .

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Step 1.1.6.4.1

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

$f' (x) = \frac{x^{2} x^{2} - 27 x^{2}}{{(x^{2} - 9)}^{2}}$

Step 1.1.6.4.2

Factor $x^{2}$ out of $- 27 x^{2}$ .

$f' (x) = \frac{x^{2} x^{2} + x^{2} \cdot - 27}{{(x^{2} - 9)}^{2}}$

Step 1.1.6.4.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 27$ .

$f' (x) = \frac{x^{2} (x^{2} - 27)}{{(x^{2} - 9)}^{2}}$

Step 1.1.6.5

Simplify the denominator.

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Step 1.1.6.5.1

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

$f' (x) = \frac{x^{2} (x^{2} - 27)}{{(x^{2} - 3^{2})}^{2}}$

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 $b = 3$ .

$f' (x) = \frac{x^{2} (x^{2} - 27)}{{((x + 3) (x - 3))}^{2}}$

Step 1.1.6.5.3

Apply the product rule to $(x + 3) (x - 3)$ .

$f' (x) = \frac{x^{2} (x^{2} - 27)}{{(x + 3)}^{2} {(x - 3)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x^{2} (x^{2} - 27)}{{(x + 3)}^{2} {(x - 3)}^{2}}$ .

$\frac{x^{2} (x^{2} - 27)}{{(x + 3)}^{2} {(x - 3)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{x^{2} (x^{2} - 27)}{{(x + 3)}^{2} {(x - 3)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{x^{2} (x^{2} - 27)}{{(x + 3)}^{2} {(x - 3)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$x^{2} (x^{2} - 27) = 0$

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 $0$ .

$x^{2} = 0$

$x^{2} - 27 = 0$

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 $x^{2} - 27$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $x^{2} - 27$ equal to $0$ .

$x^{2} - 27 = 0$

Step 2.3.3.2

Solve $x^{2} - 27 = 0$ for $x$ .

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Step 2.3.3.2.1

Add $27$ to both sides of the equation.

$x^{2} = 27$

Step 2.3.3.2.2

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

$x = \pm \sqrt{27}$

Step 2.3.3.2.3

Simplify $\pm \sqrt{27}$ .

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Step 2.3.3.2.3.1

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

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Step 2.3.3.2.3.1.1

Factor $9$ out of $27$ .

$x = \pm \sqrt{9 (3)}$

Step 2.3.3.2.3.1.2

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

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

Step 2.3.3.2.3.2

Pull terms out from under the radical.

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

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.

$x = 3 \sqrt{3}$

Step 2.3.3.2.4.2

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

$x = - 3 \sqrt{3}$

Step 2.3.3.2.4.3

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

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

Step 2.3.4

The final solution is all the values that make $x^{2} (x^{2} - 27) = 0$ true.

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

Step 3

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

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

Step 4

Find where the derivative is undefined.

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Step 4.1

Set the denominator in $\frac{x^{2} (x^{2} - 27)}{{(x + 3)}^{2} {(x - 3)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

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 $0$ .

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

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

Step 4.2.2

Set ${(x + 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.2.2.1

Set ${(x + 3)}^{2}$ equal to $0$ .

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

Step 4.2.2.2

Solve ${(x + 3)}^{2} = 0$ for $x$ .

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Step 4.2.2.2.1

Set the $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 4.2.2.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4.2.3

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.2.3.1

Set ${(x - 3)}^{2}$ equal to $0$ .

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

Step 4.2.3.2

Solve ${(x - 3)}^{2} = 0$ for $x$ .

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Step 4.2.3.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.2.3.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4.2.4

The final solution is all the values that make ${(x + 3)}^{2} {(x - 3)}^{2} = 0$ true.

$x = - 3, 3$

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 $0$ .

$x = - 3, x = 3$

Step 5

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

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

Step 6

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

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Step 6.1

Replace the variable $x$ with $- 6.196152$ in the expression.

$f' (- 6.196152) = \frac{{(- 6.196152)}^{2} ({(- 6.196152)}^{2} - 27)}{{((- 6.196152) + 3)}^{2} {((- 6.196152) - 3)}^{2}}$

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 $- 6.196152$ to the power of $2$ .

$f' (- 6.196152) = \frac{{(- 6.196152)}^{2} (38.3922996 - 27)}{{(- 6.196152 + 3)}^{2} {(- 6.196152 - 3)}^{2}}$

Step 6.2.1.2

Subtract $27$ from $38.3922996$ .

$f' (- 6.196152) = \frac{{(- 6.196152)}^{2} \cdot 11.3922996}{{(- 6.196152 + 3)}^{2} {(- 6.196152 - 3)}^{2}}$

Step 6.2.1.3

Raise $- 6.196152$ to the power of $2$ .

$f' (- 6.196152) = \frac{38.3922996 \cdot 11.3922996}{{(- 6.196152 + 3)}^{2} {(- 6.196152 - 3)}^{2}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Add $- 6.196152$ and $3$ .

$f' (- 6.196152) = \frac{38.3922996 \cdot 11.3922996}{{(- 3.196152)}^{2} {(- 6.196152 - 3)}^{2}}$

Step 6.2.2.2

Subtract $3$ from $- 6.196152$ .

$f' (- 6.196152) = \frac{38.3922996 \cdot 11.3922996}{{(- 3.196152)}^{2} {(- 9.196152)}^{2}}$

Step 6.2.2.3

Raise $- 3.196152$ to the power of $2$ .

$f' (- 6.196152) = \frac{38.3922996 \cdot 11.3922996}{10.2153876 {(- 9.196152)}^{2}}$

Step 6.2.2.4

Raise $- 9.196152$ to the power of $2$ .

$f' (- 6.196152) = \frac{38.3922996 \cdot 11.3922996}{10.2153876 \cdot 84.5692116}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Multiply $38.3922996$ by $11.3922996$ .

$f' (- 6.196152) = \frac{437.37657972}{10.2153876 \cdot 84.5692116}$

Step 6.2.3.2

Multiply $10.2153876$ by $84.5692116$ .

$f' (- 6.196152) = \frac{437.37657972}{863.90727619}$

Step 6.2.3.3

Divide $437.37657972$ by $863.90727619$ .

$f' (- 6.196152) = 0.50627722$

Step 6.2.4

The final answer is $0.50627722$ .

$0.50627722$

Step 6.3

At $x = - 6.196152$ the derivative is $0.50627722$ . Since this is positive, the function is increasing on $(- \infty, - 3 \sqrt{3})$ .

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

Step 7

Substitute a value from the interval $(- 5.196152, - 3)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 4.098076$ in the expression.

$f' (- 4.098076) = \frac{{(- 4.098076)}^{2} ({(- 4.098076)}^{2} - 27)}{{((- 4.098076) + 3)}^{2} {((- 4.098076) - 3)}^{2}}$

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 $- 4.098076$ to the power of $2$ .

$f' (- 4.098076) = \frac{{(- 4.098076)}^{2} (16.7942269 - 27)}{{(- 4.098076 + 3)}^{2} {(- 4.098076 - 3)}^{2}}$

Step 7.2.1.2

Subtract $27$ from $16.7942269$ .

$f' (- 4.098076) = \frac{{(- 4.098076)}^{2} \cdot - 10.20577309}{{(- 4.098076 + 3)}^{2} {(- 4.098076 - 3)}^{2}}$

Step 7.2.1.3

Raise $- 4.098076$ to the power of $2$ .

$f' (- 4.098076) = \frac{16.7942269 \cdot - 10.20577309}{{(- 4.098076 + 3)}^{2} {(- 4.098076 - 3)}^{2}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Add $- 4.098076$ and $3$ .

$f' (- 4.098076) = \frac{16.7942269 \cdot - 10.20577309}{{(- 1.098076)}^{2} {(- 4.098076 - 3)}^{2}}$

Step 7.2.2.2

Subtract $3$ from $- 4.098076$ .

$f' (- 4.098076) = \frac{16.7942269 \cdot - 10.20577309}{{(- 1.098076)}^{2} {(- 7.098076)}^{2}}$

Step 7.2.2.3

Raise $- 1.098076$ to the power of $2$ .

$f' (- 4.098076) = \frac{16.7942269 \cdot - 10.20577309}{1.2057709 {(- 7.098076)}^{2}}$

Step 7.2.2.4

Raise $- 7.098076$ to the power of $2$ .

$f' (- 4.098076) = \frac{16.7942269 \cdot - 10.20577309}{1.2057709 \cdot 50.3826829}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Multiply $16.7942269$ by $- 10.20577309$ .

$f' (- 4.098076) = \frac{- 171.39806911}{1.2057709 \cdot 50.3826829}$

Step 7.2.3.2

Multiply $1.2057709$ by $50.3826829$ .

$f' (- 4.098076) = \frac{- 171.39806911}{60.74997299}$

Step 7.2.3.3

Divide $- 171.39806911$ by $60.74997299$ .

$f' (- 4.098076) = - 2.82136864$

Step 7.2.4

The final answer is $- 2.82136864$ .

$- 2.82136864$

Step 7.3

At $x = - 4.098076$ the derivative is $- 2.82136864$ . Since this is negative, the function is decreasing on $(- 5.196152, - 3)$ .

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

Step 8

Substitute a value from the interval $(- 3, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $- \frac{3}{2}$ in the expression.

$f' (- \frac{3}{2}) = \frac{{(- \frac{3}{2})}^{2} ({(- \frac{3}{2})}^{2} - 27)}{{((- \frac{3}{2}) + 3)}^{2} {((- \frac{3}{2}) - 3)}^{2}}$

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

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

$f' (- \frac{3}{2}) = \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2} ({(- \frac{3}{2})}^{2} - 27)}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.2.1.2.1

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

$f' (- \frac{3}{2}) = \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2} ({(- 1)}^{2} {(\frac{3}{2})}^{2} - 27)}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.2.2

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

$f' (- \frac{3}{2}) = \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2} ({(- 1)}^{2} (\frac{3^{2}}{2^{2}}) - 27)}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.3

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

$f' (- \frac{3}{2}) = \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2} (1 (\frac{3^{2}}{2^{2}}) - 27)}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.4

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$f' (- \frac{3}{2}) = \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2} (\frac{3^{2}}{2^{2}} - 27)}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.5

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

$f' (- \frac{3}{2}) = \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2} (\frac{9}{2^{2}} - 27)}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.6

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

$f' (- \frac{3}{2}) = \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2} (\frac{9}{4} - 27)}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.7

To write $- 27$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f' (- \frac{3}{2}) = \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2} (\frac{9}{4} - 27 \cdot \frac{4}{4})}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.8

Combine $- 27$ and $\frac{4}{4}$ .

$f' (- \frac{3}{2}) = \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2} (\frac{9}{4} + \frac{- 27 \cdot 4}{4})}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.9

Combine the numerators over the common denominator.

$f' (- \frac{3}{2}) = \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2} (\frac{9 - 27 \cdot 4}{4})}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.10

Simplify the numerator.

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Step 8.2.1.10.1

Multiply $- 27$ by $4$ .

$f' (- \frac{3}{2}) = \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2} (\frac{9 - 108}{4})}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.10.2

Subtract $108$ from $9$ .

$f' (- \frac{3}{2}) = \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2} (\frac{- 99}{4})}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.11

Move the negative in front of the fraction.

$f' (- \frac{3}{2}) = \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2} \cdot (- 1 (\frac{99}{4}))}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.12

Combine exponents.

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Step 8.2.1.12.1

Factor out negative.

$f' (- \frac{3}{2}) = \frac{- ({(- 1)}^{2} {(\frac{3}{2})}^{2} (\frac{99}{4}))}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.12.2

Combine $\frac{99}{4}$ and ${(- 1)}^{2}$ .

$f' (- \frac{3}{2}) = \frac{- (\frac{99 {(- 1)}^{2}}{4} \cdot {(\frac{3}{2})}^{2})}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.12.3

Combine $\frac{99 {(- 1)}^{2}}{4}$ and ${(\frac{3}{2})}^{2}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{99 {(- 1)}^{2} {(\frac{3}{2})}^{2}}{4}}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.13

Simplify the numerator.

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Step 8.2.1.13.1

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

$f' (- \frac{3}{2}) = \frac{- \frac{99 {(- 1)}^{2} (\frac{3^{2}}{2^{2}})}{4}}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.13.2

Combine exponents.

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Step 8.2.1.13.2.1

Combine $99$ and $\frac{3^{2}}{2^{2}}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{{(- 1)}^{2} (\frac{99 \cdot 3^{2}}{2^{2}})}{4}}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.13.2.2

Combine ${(- 1)}^{2}$ and $\frac{99 \cdot 3^{2}}{2^{2}}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{\frac{{(- 1)}^{2} (99 \cdot 3^{2})}{2^{2}}}{4}}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.13.3

Simplify the numerator.

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Step 8.2.1.13.3.1

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

$f' (- \frac{3}{2}) = \frac{- \frac{\frac{1 \cdot 99 \cdot 3^{2}}{2^{2}}}{4}}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.13.3.2

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

$f' (- \frac{3}{2}) = \frac{- \frac{\frac{1 \cdot 99 \cdot 9}{2^{2}}}{4}}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.13.3.3

Combine exponents.

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Step 8.2.1.13.3.3.1

Multiply $99$ by $1$ .

$f' (- \frac{3}{2}) = \frac{- \frac{\frac{99 \cdot 9}{2^{2}}}{4}}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.13.3.3.2

Multiply $99$ by $9$ .

$f' (- \frac{3}{2}) = \frac{- \frac{\frac{891}{2^{2}}}{4}}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.13.4

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

$f' (- \frac{3}{2}) = \frac{- \frac{\frac{891}{4}}{4}}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.14

Multiply the numerator by the reciprocal of the denominator.

$f' (- \frac{3}{2}) = \frac{- (\frac{891}{4} \cdot \frac{1}{4})}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.15

Multiply $\frac{891}{4} \cdot \frac{1}{4}$ .

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Step 8.2.1.15.1

Multiply $\frac{891}{4}$ by $\frac{1}{4}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{4 \cdot 4}}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.1.15.2

Multiply $4$ by $4$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{{(- \frac{3}{2} + 3)}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{{(- \frac{3}{2} + 3 \cdot \frac{2}{2})}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.2.2

Combine $3$ and $\frac{2}{2}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{{(- \frac{3}{2} + \frac{3 \cdot 2}{2})}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.2.3

Combine the numerators over the common denominator.

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{{(\frac{- 3 + 3 \cdot 2}{2})}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.2.4

Simplify the numerator.

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Step 8.2.2.4.1

Multiply $3$ by $2$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{{(\frac{- 3 + 6}{2})}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.2.4.2

Add $- 3$ and $6$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{{(\frac{3}{2})}^{2} {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.2.5

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

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{3^{2}}{2^{2}} \cdot {(- \frac{3}{2} - 3)}^{2}}$

Step 8.2.2.6

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{3^{2}}{2^{2}} \cdot {(- \frac{3}{2} - 3 \cdot \frac{2}{2})}^{2}}$

Step 8.2.2.7

Combine $- 3$ and $\frac{2}{2}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{3^{2}}{2^{2}} \cdot {(- \frac{3}{2} + \frac{- 3 \cdot 2}{2})}^{2}}$

Step 8.2.2.8

Combine the numerators over the common denominator.

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{3^{2}}{2^{2}} \cdot {(\frac{- 3 - 3 \cdot 2}{2})}^{2}}$

Step 8.2.2.9

Simplify the numerator.

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Step 8.2.2.9.1

Multiply $- 3$ by $2$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{3^{2}}{2^{2}} \cdot {(\frac{- 3 - 6}{2})}^{2}}$

Step 8.2.2.9.2

Subtract $6$ from $- 3$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{3^{2}}{2^{2}} \cdot {(\frac{- 9}{2})}^{2}}$

Step 8.2.2.10

Move the negative in front of the fraction.

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{3^{2}}{2^{2}} \cdot {(- \frac{9}{2})}^{2}}$

Step 8.2.2.11

Apply the product rule to $- \frac{9}{2}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{3^{2}}{2^{2}} \cdot ({(- 1)}^{2} {(\frac{9}{2})}^{2})}$

Step 8.2.2.12

Combine exponents.

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Step 8.2.2.12.1

Combine $\frac{3^{2}}{2^{2}}$ and ${(- 1)}^{2}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{3^{2} {(- 1)}^{2}}{2^{2}} \cdot {(\frac{9}{2})}^{2}}$

Step 8.2.2.12.2

Combine $\frac{3^{2} {(- 1)}^{2}}{2^{2}}$ and ${(\frac{9}{2})}^{2}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{3^{2} {(- 1)}^{2} {(\frac{9}{2})}^{2}}{2^{2}}}$

Step 8.2.2.13

Simplify the numerator.

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Step 8.2.2.13.1

Apply the product rule to $\frac{9}{2}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{3^{2} {(- 1)}^{2} (\frac{9^{2}}{2^{2}})}{2^{2}}}$

Step 8.2.2.13.2

Combine exponents.

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Step 8.2.2.13.2.1

Combine $3^{2}$ and $\frac{9^{2}}{2^{2}}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{{(- 1)}^{2} (\frac{3^{2} \cdot 9^{2}}{2^{2}})}{2^{2}}}$

Step 8.2.2.13.2.2

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

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{{(- 1)}^{2} (\frac{3^{2} \cdot {(3^{2})}^{2}}{2^{2}})}{2^{2}}}$

Step 8.2.2.13.2.3

Multiply the exponents in ${(3^{2})}^{2}$ .

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Step 8.2.2.13.2.3.1

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

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{{(- 1)}^{2} (\frac{3^{2} \cdot 3^{2 \cdot 2}}{2^{2}})}{2^{2}}}$

Step 8.2.2.13.2.3.2

Multiply $2$ by $2$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{{(- 1)}^{2} (\frac{3^{2} \cdot 3^{4}}{2^{2}})}{2^{2}}}$

Step 8.2.2.13.2.4

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

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{{(- 1)}^{2} (\frac{3^{2 + 4}}{2^{2}})}{2^{2}}}$

Step 8.2.2.13.2.5

Add $2$ and $4$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{{(- 1)}^{2} (\frac{3^{6}}{2^{2}})}{2^{2}}}$

Step 8.2.2.13.2.6

Combine ${(- 1)}^{2}$ and $\frac{3^{6}}{2^{2}}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{\frac{{(- 1)}^{2} \cdot 3^{6}}{2^{2}}}{2^{2}}}$

Step 8.2.2.13.3

Simplify the numerator.

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Step 8.2.2.13.3.1

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

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{\frac{1 \cdot 3^{6}}{2^{2}}}{2^{2}}}$

Step 8.2.2.13.3.2

Raise $3$ to the power of $6$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{\frac{1 \cdot 729}{2^{2}}}{2^{2}}}$

Step 8.2.2.13.3.3

Multiply $729$ by $1$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{\frac{729}{2^{2}}}{2^{2}}}$

Step 8.2.2.13.4

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

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{\frac{729}{4}}{2^{2}}}$

Step 8.2.2.14

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

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{\frac{729}{4}}{4}}$

Step 8.2.2.15

Multiply the numerator by the reciprocal of the denominator.

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{729}{4} \cdot \frac{1}{4}}$

Step 8.2.2.16

Multiply $\frac{729}{4} \cdot \frac{1}{4}$ .

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Step 8.2.2.16.1

Multiply $\frac{729}{4}$ by $\frac{1}{4}$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{729}{4 \cdot 4}}$

Step 8.2.2.16.2

Multiply $4$ by $4$ .

$f' (- \frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{729}{16}}$

Step 8.2.3

Multiply the numerator by the reciprocal of the denominator.

$f' (- \frac{3}{2}) = - \frac{891}{16} \cdot \frac{16}{729}$

Step 8.2.4

Cancel the common factor of $81$ .

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Step 8.2.4.1

Move the leading negative in $- \frac{891}{16}$ into the numerator.

$f' (- \frac{3}{2}) = \frac{- 891}{16} \cdot \frac{16}{729}$

Step 8.2.4.2

Factor $81$ out of $- 891$ .

$f' (- \frac{3}{2}) = \frac{81 (- 11)}{16} \cdot \frac{16}{729}$

Step 8.2.4.3

Factor $81$ out of $729$ .

$f' (- \frac{3}{2}) = \frac{81 \cdot - 11}{16} \cdot \frac{16}{81 \cdot 9}$

Step 8.2.4.4

Cancel the common factor.

$f' (- \frac{3}{2}) = \frac{81 \cdot - 11}{16} \cdot \frac{16}{81 \cdot 9}$

Step 8.2.4.5

Rewrite the expression.

$f' (- \frac{3}{2}) = \frac{- 11}{16} \cdot \frac{16}{9}$

Step 8.2.5

Cancel the common factor of $16$ .

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Step 8.2.5.1

Cancel the common factor.

$f' (- \frac{3}{2}) = \frac{- 11}{16} \cdot \frac{16}{9}$

Step 8.2.5.2

Rewrite the expression.

$f' (- \frac{3}{2}) = - 11 (\frac{1}{9})$

Step 8.2.6

Combine $- 11$ and $\frac{1}{9}$ .

$f' (- \frac{3}{2}) = \frac{- 11}{9}$

Step 8.2.7

Move the negative in front of the fraction.

$f' (- \frac{3}{2}) = - \frac{11}{9}$

Step 8.2.8

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

$- \frac{11}{9}$

Step 8.3

At $x = - \frac{3}{2}$ the derivative is $- \frac{11}{9}$ . Since this is negative, the function is decreasing on $(- 3, 0)$ .

Decreasing on $(- 3, 0)$ since $f' (x) < 0$

Step 9

Substitute a value from the interval $(0, 3)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $\frac{3}{2}$ in the expression.

$f' (\frac{3}{2}) = \frac{{(\frac{3}{2})}^{2} ({(\frac{3}{2})}^{2} - 27)}{{((\frac{3}{2}) + 3)}^{2} {((\frac{3}{2}) - 3)}^{2}}$

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

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

$f' (\frac{3}{2}) = \frac{\frac{3^{2}}{2^{2}} \cdot ({(\frac{3}{2})}^{2} - 27)}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.2

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

$f' (\frac{3}{2}) = \frac{\frac{3^{2}}{2^{2}} \cdot (\frac{3^{2}}{2^{2}} - 27)}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.3

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

$f' (\frac{3}{2}) = \frac{\frac{3^{2}}{2^{2}} \cdot (\frac{9}{2^{2}} - 27)}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.4

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

$f' (\frac{3}{2}) = \frac{\frac{3^{2}}{2^{2}} \cdot (\frac{9}{4} - 27)}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.5

To write $- 27$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f' (\frac{3}{2}) = \frac{\frac{3^{2}}{2^{2}} \cdot (\frac{9}{4} - 27 \cdot \frac{4}{4})}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.6

Combine $- 27$ and $\frac{4}{4}$ .

$f' (\frac{3}{2}) = \frac{\frac{3^{2}}{2^{2}} \cdot (\frac{9}{4} + \frac{- 27 \cdot 4}{4})}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.7

Combine the numerators over the common denominator.

$f' (\frac{3}{2}) = \frac{\frac{3^{2}}{2^{2}} \cdot \frac{9 - 27 \cdot 4}{4}}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.8

Simplify the numerator.

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Step 9.2.1.8.1

Multiply $- 27$ by $4$ .

$f' (\frac{3}{2}) = \frac{\frac{3^{2}}{2^{2}} \cdot \frac{9 - 108}{4}}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.8.2

Subtract $108$ from $9$ .

$f' (\frac{3}{2}) = \frac{\frac{3^{2}}{2^{2}} \cdot \frac{- 99}{4}}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.9

Move the negative in front of the fraction.

$f' (\frac{3}{2}) = \frac{\frac{3^{2}}{2^{2}} \cdot (- 1 (\frac{99}{4}))}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.10

Combine exponents.

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Step 9.2.1.10.1

Factor out negative.

$f' (\frac{3}{2}) = \frac{- (\frac{3^{2}}{2^{2}} \cdot \frac{99}{4})}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.10.2

Multiply $\frac{3^{2}}{2^{2}}$ by $\frac{99}{4}$ .

$f' (\frac{3}{2}) = \frac{- \frac{3^{2} \cdot 99}{2^{2} \cdot 4}}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.10.3

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

$f' (\frac{3}{2}) = \frac{- \frac{3^{2} \cdot 99}{2^{2} \cdot 2^{2}}}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.10.4

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

$f' (\frac{3}{2}) = \frac{- \frac{3^{2} \cdot 99}{2^{2 + 2}}}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.10.5

Add $2$ and $2$ .

$f' (\frac{3}{2}) = \frac{- \frac{3^{2} \cdot 99}{2^{4}}}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.11

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

$f' (\frac{3}{2}) = \frac{- \frac{9 \cdot 99}{2^{4}}}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.12

Raise $2$ to the power of $4$ .

$f' (\frac{3}{2}) = \frac{- \frac{9 \cdot 99}{16}}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.1.13

Multiply $9$ by $99$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{{(\frac{3}{2} + 3)}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{{(\frac{3}{2} + 3 \cdot \frac{2}{2})}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.2.2

Combine $3$ and $\frac{2}{2}$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{{(\frac{3}{2} + \frac{3 \cdot 2}{2})}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.2.3

Combine the numerators over the common denominator.

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{{(\frac{3 + 3 \cdot 2}{2})}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.2.4

Simplify the numerator.

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Step 9.2.2.4.1

Multiply $3$ by $2$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{{(\frac{3 + 6}{2})}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.2.4.2

Add $3$ and $6$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{{(\frac{9}{2})}^{2} {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.2.5

Apply the product rule to $\frac{9}{2}$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{9^{2}}{2^{2}} \cdot {(\frac{3}{2} - 3)}^{2}}$

Step 9.2.2.6

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{9^{2}}{2^{2}} \cdot {(\frac{3}{2} - 3 \cdot \frac{2}{2})}^{2}}$

Step 9.2.2.7

Combine $- 3$ and $\frac{2}{2}$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{9^{2}}{2^{2}} \cdot {(\frac{3}{2} + \frac{- 3 \cdot 2}{2})}^{2}}$

Step 9.2.2.8

Combine the numerators over the common denominator.

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{9^{2}}{2^{2}} \cdot {(\frac{3 - 3 \cdot 2}{2})}^{2}}$

Step 9.2.2.9

Simplify the numerator.

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Step 9.2.2.9.1

Multiply $- 3$ by $2$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{9^{2}}{2^{2}} \cdot {(\frac{3 - 6}{2})}^{2}}$

Step 9.2.2.9.2

Subtract $6$ from $3$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{9^{2}}{2^{2}} \cdot {(\frac{- 3}{2})}^{2}}$

Step 9.2.2.10

Move the negative in front of the fraction.

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{9^{2}}{2^{2}} \cdot {(- \frac{3}{2})}^{2}}$

Step 9.2.2.11

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

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{9^{2}}{2^{2}} \cdot ({(- 1)}^{2} {(\frac{3}{2})}^{2})}$

Step 9.2.2.12

Combine exponents.

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Step 9.2.2.12.1

Combine $\frac{9^{2}}{2^{2}}$ and ${(- 1)}^{2}$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{9^{2} {(- 1)}^{2}}{2^{2}} \cdot {(\frac{3}{2})}^{2}}$

Step 9.2.2.12.2

Combine $\frac{9^{2} {(- 1)}^{2}}{2^{2}}$ and ${(\frac{3}{2})}^{2}$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{9^{2} {(- 1)}^{2} {(\frac{3}{2})}^{2}}{2^{2}}}$

Step 9.2.2.13

Simplify the numerator.

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Step 9.2.2.13.1

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

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{9^{2} {(- 1)}^{2} (\frac{3^{2}}{2^{2}})}{2^{2}}}$

Step 9.2.2.13.2

Combine exponents.

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Step 9.2.2.13.2.1

Combine $9^{2}$ and $\frac{3^{2}}{2^{2}}$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{{(- 1)}^{2} (\frac{9^{2} \cdot 3^{2}}{2^{2}})}{2^{2}}}$

Step 9.2.2.13.2.2

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

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{{(- 1)}^{2} (\frac{{(3^{2})}^{2} \cdot 3^{2}}{2^{2}})}{2^{2}}}$

Step 9.2.2.13.2.3

Multiply the exponents in ${(3^{2})}^{2}$ .

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Step 9.2.2.13.2.3.1

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

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{{(- 1)}^{2} (\frac{3^{2 \cdot 2} \cdot 3^{2}}{2^{2}})}{2^{2}}}$

Step 9.2.2.13.2.3.2

Multiply $2$ by $2$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{{(- 1)}^{2} (\frac{3^{4} \cdot 3^{2}}{2^{2}})}{2^{2}}}$

Step 9.2.2.13.2.4

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

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{{(- 1)}^{2} (\frac{3^{4 + 2}}{2^{2}})}{2^{2}}}$

Step 9.2.2.13.2.5

Add $4$ and $2$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{{(- 1)}^{2} (\frac{3^{6}}{2^{2}})}{2^{2}}}$

Step 9.2.2.13.2.6

Combine ${(- 1)}^{2}$ and $\frac{3^{6}}{2^{2}}$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{\frac{{(- 1)}^{2} \cdot 3^{6}}{2^{2}}}{2^{2}}}$

Step 9.2.2.13.3

Simplify the numerator.

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Step 9.2.2.13.3.1

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

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{\frac{1 \cdot 3^{6}}{2^{2}}}{2^{2}}}$

Step 9.2.2.13.3.2

Raise $3$ to the power of $6$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{\frac{1 \cdot 729}{2^{2}}}{2^{2}}}$

Step 9.2.2.13.3.3

Multiply $729$ by $1$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{\frac{729}{2^{2}}}{2^{2}}}$

Step 9.2.2.13.4

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

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{\frac{729}{4}}{2^{2}}}$

Step 9.2.2.14

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

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{\frac{729}{4}}{4}}$

Step 9.2.2.15

Multiply the numerator by the reciprocal of the denominator.

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{729}{4} \cdot \frac{1}{4}}$

Step 9.2.2.16

Multiply $\frac{729}{4} \cdot \frac{1}{4}$ .

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Step 9.2.2.16.1

Multiply $\frac{729}{4}$ by $\frac{1}{4}$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{729}{4 \cdot 4}}$

Step 9.2.2.16.2

Multiply $4$ by $4$ .

$f' (\frac{3}{2}) = \frac{- \frac{891}{16}}{\frac{729}{16}}$

Step 9.2.3

Multiply the numerator by the reciprocal of the denominator.

$f' (\frac{3}{2}) = - \frac{891}{16} \cdot \frac{16}{729}$

Step 9.2.4

Cancel the common factor of $81$ .

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Step 9.2.4.1

Move the leading negative in $- \frac{891}{16}$ into the numerator.

$f' (\frac{3}{2}) = \frac{- 891}{16} \cdot \frac{16}{729}$

Step 9.2.4.2

Factor $81$ out of $- 891$ .

$f' (\frac{3}{2}) = \frac{81 (- 11)}{16} \cdot \frac{16}{729}$

Step 9.2.4.3

Factor $81$ out of $729$ .

$f' (\frac{3}{2}) = \frac{81 \cdot - 11}{16} \cdot \frac{16}{81 \cdot 9}$

Step 9.2.4.4

Cancel the common factor.

$f' (\frac{3}{2}) = \frac{81 \cdot - 11}{16} \cdot \frac{16}{81 \cdot 9}$

Step 9.2.4.5

Rewrite the expression.

$f' (\frac{3}{2}) = \frac{- 11}{16} \cdot \frac{16}{9}$

Step 9.2.5

Cancel the common factor of $16$ .

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Step 9.2.5.1

Cancel the common factor.

$f' (\frac{3}{2}) = \frac{- 11}{16} \cdot \frac{16}{9}$

Step 9.2.5.2

Rewrite the expression.

$f' (\frac{3}{2}) = - 11 (\frac{1}{9})$

Step 9.2.6

Combine $- 11$ and $\frac{1}{9}$ .

$f' (\frac{3}{2}) = \frac{- 11}{9}$

Step 9.2.7

Move the negative in front of the fraction.

$f' (\frac{3}{2}) = - \frac{11}{9}$

Step 9.2.8

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

$- \frac{11}{9}$

Step 9.3

At $x = \frac{3}{2}$ the derivative is $- \frac{11}{9}$ . Since this is negative, the function is decreasing on $(0, 3)$ .

Decreasing on $(0, 3)$ since $f' (x) < 0$

Step 10

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

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Step 10.1

Replace the variable $x$ with $4.098076$ in the expression.

$f' (4.098076) = \frac{{(4.098076)}^{2} ({(4.098076)}^{2} - 27)}{{((4.098076) + 3)}^{2} {((4.098076) - 3)}^{2}}$

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 $4.098076$ to the power of $2$ .

$f' (4.098076) = \frac{{4.098076}^{2} (16.7942269 - 27)}{{(4.098076 + 3)}^{2} {(4.098076 - 3)}^{2}}$

Step 10.2.1.2

Subtract $27$ from $16.7942269$ .

$f' (4.098076) = \frac{{4.098076}^{2} \cdot - 10.20577309}{{(4.098076 + 3)}^{2} {(4.098076 - 3)}^{2}}$

Step 10.2.1.3

Raise $4.098076$ to the power of $2$ .

$f' (4.098076) = \frac{16.7942269 \cdot - 10.20577309}{{(4.098076 + 3)}^{2} {(4.098076 - 3)}^{2}}$

Step 10.2.2

Simplify the denominator.

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Step 10.2.2.1

Add $4.098076$ and $3$ .

$f' (4.098076) = \frac{16.7942269 \cdot - 10.20577309}{{7.098076}^{2} {(4.098076 - 3)}^{2}}$

Step 10.2.2.2

Subtract $3$ from $4.098076$ .

$f' (4.098076) = \frac{16.7942269 \cdot - 10.20577309}{{7.098076}^{2} \cdot {1.098076}^{2}}$

Step 10.2.2.3

Raise $7.098076$ to the power of $2$ .

$f' (4.098076) = \frac{16.7942269 \cdot - 10.20577309}{50.3826829 \cdot {1.098076}^{2}}$

Step 10.2.2.4

Raise $1.098076$ to the power of $2$ .

$f' (4.098076) = \frac{16.7942269 \cdot - 10.20577309}{50.3826829 \cdot 1.2057709}$

Step 10.2.3

Simplify the expression.

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Step 10.2.3.1

Multiply $16.7942269$ by $- 10.20577309$ .

$f' (4.098076) = \frac{- 171.39806911}{50.3826829 \cdot 1.2057709}$

Step 10.2.3.2

Multiply $50.3826829$ by $1.2057709$ .

$f' (4.098076) = \frac{- 171.39806911}{60.74997299}$

Step 10.2.3.3

Divide $- 171.39806911$ by $60.74997299$ .

$f' (4.098076) = - 2.82136864$

Step 10.2.4

The final answer is $- 2.82136864$ .

$- 2.82136864$

Step 10.3

At $x = 4.098076$ the derivative is $- 2.82136864$ . Since this is negative, the function is decreasing on $(3, 3 \sqrt{3})$ .

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

Step 11

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

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Step 11.1

Replace the variable $x$ with $6.196152$ in the expression.

$f' (6.196152) = \frac{{(6.196152)}^{2} ({(6.196152)}^{2} - 27)}{{((6.196152) + 3)}^{2} {((6.196152) - 3)}^{2}}$

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 $6.196152$ to the power of $2$ .

$f' (6.196152) = \frac{{6.196152}^{2} (38.3922996 - 27)}{{(6.196152 + 3)}^{2} {(6.196152 - 3)}^{2}}$

Step 11.2.1.2

Subtract $27$ from $38.3922996$ .

$f' (6.196152) = \frac{{6.196152}^{2} \cdot 11.3922996}{{(6.196152 + 3)}^{2} {(6.196152 - 3)}^{2}}$

Step 11.2.1.3

Raise $6.196152$ to the power of $2$ .

$f' (6.196152) = \frac{38.3922996 \cdot 11.3922996}{{(6.196152 + 3)}^{2} {(6.196152 - 3)}^{2}}$

Step 11.2.2

Simplify the denominator.

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Step 11.2.2.1

Add $6.196152$ and $3$ .

$f' (6.196152) = \frac{38.3922996 \cdot 11.3922996}{{9.196152}^{2} {(6.196152 - 3)}^{2}}$

Step 11.2.2.2

Subtract $3$ from $6.196152$ .

$f' (6.196152) = \frac{38.3922996 \cdot 11.3922996}{{9.196152}^{2} \cdot {3.196152}^{2}}$

Step 11.2.2.3

Raise $9.196152$ to the power of $2$ .

$f' (6.196152) = \frac{38.3922996 \cdot 11.3922996}{84.5692116 \cdot {3.196152}^{2}}$

Step 11.2.2.4

Raise $3.196152$ to the power of $2$ .

$f' (6.196152) = \frac{38.3922996 \cdot 11.3922996}{84.5692116 \cdot 10.2153876}$

Step 11.2.3

Simplify the expression.

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Step 11.2.3.1

Multiply $38.3922996$ by $11.3922996$ .

$f' (6.196152) = \frac{437.37657972}{84.5692116 \cdot 10.2153876}$

Step 11.2.3.2

Multiply $84.5692116$ by $10.2153876$ .

$f' (6.196152) = \frac{437.37657972}{863.90727619}$

Step 11.2.3.3

Divide $437.37657972$ by $863.90727619$ .

$f' (6.196152) = 0.50627722$

Step 11.2.4

The final answer is $0.50627722$ .

$0.50627722$

Step 11.3

At $x = 6.196152$ the derivative is $0.50627722$ . Since this is positive, the function is increasing on $(3 \sqrt{3}, \infty)$ .

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

Step 12

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

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

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

Step 13