Calculus Examples

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

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.

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Step 1.1.1.1

By the Sum Rule, the derivative of $1 + \frac{1}{x} + \frac{9}{x^{2}} + \frac{1}{x^{3}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [\frac{9}{x^{2}}] + \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [\frac{9}{x^{2}}] + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [\frac{9}{x^{2}}] + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{x}]$ .

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Step 1.1.2.1

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$0 + \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [\frac{9}{x^{2}}] + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.2.2

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

$0 - x^{- 2} + \frac{d}{d x} [\frac{9}{x^{2}}] + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.3

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

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Step 1.1.3.1

Since $9$ is constant with respect to $x$ , the derivative of $\frac{9}{x^{2}}$ with respect to $x$ is $9 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$0 - x^{- 2} + 9 \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.3.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$0 - x^{- 2} + 9 \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 1.1.3.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

$0 - x^{- 2} + 9 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.3.3.2

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

$0 - x^{- 2} + 9 (- {u_{1}}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.3.3.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$0 - x^{- 2} + 9 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.3.4

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

$0 - x^{- 2} + 9 (- {(x^{2})}^{- 2} (2 x)) + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.3.5

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

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Step 1.1.3.5.1

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

$0 - x^{- 2} + 9 (- x^{2 \cdot - 2} (2 x)) + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.3.5.2

Multiply $2$ by $- 2$ .

$0 - x^{- 2} + 9 (- x^{- 4} (2 x)) + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.3.6

Multiply $2$ by $- 1$ .

$0 - x^{- 2} + 9 (- 2 x^{- 4} x) + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.3.7

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

$0 - x^{- 2} + 9 (- 2 (x^{1} x^{- 4})) + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.3.8

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

$0 - x^{- 2} + 9 (- 2 x^{1 - 4}) + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.3.9

Subtract $4$ from $1$ .

$0 - x^{- 2} + 9 (- 2 x^{- 3}) + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.3.10

Multiply $- 2$ by $9$ .

$0 - x^{- 2} - 18 x^{- 3} + \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.1.4

Evaluate $\frac{d}{d x} [\frac{1}{x^{3}}]$ .

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Step 1.1.4.1

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$0 - x^{- 2} - 18 x^{- 3} + \frac{d}{d x} [{(x^{3})}^{- 1}]$

Step 1.1.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{3}$ .

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Step 1.1.4.2.1

To apply the Chain Rule, set $u_{2}$ as $x^{3}$ .

$0 - x^{- 2} - 18 x^{- 3} + \frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{3}]$

Step 1.1.4.2.2

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

$0 - x^{- 2} - 18 x^{- 3} - {u_{2}}^{- 2} \frac{d}{d x} [x^{3}]$

Step 1.1.4.2.3

Replace all occurrences of $u_{2}$ with $x^{3}$ .

$0 - x^{- 2} - 18 x^{- 3} - {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}]$

Step 1.1.4.3

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

$0 - x^{- 2} - 18 x^{- 3} - {(x^{3})}^{- 2} (3 x^{2})$

Step 1.1.4.4

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

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Step 1.1.4.4.1

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

$0 - x^{- 2} - 18 x^{- 3} - x^{3 \cdot - 2} (3 x^{2})$

Step 1.1.4.4.2

Multiply $3$ by $- 2$ .

$0 - x^{- 2} - 18 x^{- 3} - x^{- 6} (3 x^{2})$

Step 1.1.4.5

Multiply $3$ by $- 1$ .

$0 - x^{- 2} - 18 x^{- 3} - 3 x^{- 6} x^{2}$

Step 1.1.4.6

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 1.1.4.6.1

Move $x^{2}$ .

$0 - x^{- 2} - 18 x^{- 3} - 3 (x^{2} x^{- 6})$

Step 1.1.4.6.2

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

$0 - x^{- 2} - 18 x^{- 3} - 3 x^{2 - 6}$

Step 1.1.4.6.3

Subtract $6$ from $2$ .

$0 - x^{- 2} - 18 x^{- 3} - 3 x^{- 4}$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 - \frac{1}{x^{2}} - 18 x^{- 3} - 3 x^{- 4}$

Step 1.1.5.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 - \frac{1}{x^{2}} - 18 \frac{1}{x^{3}} - 3 x^{- 4}$

Step 1.1.5.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 - \frac{1}{x^{2}} - 18 \frac{1}{x^{3}} - 3 \frac{1}{x^{4}}$

Step 1.1.5.4

Combine terms.

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Step 1.1.5.4.1

Subtract $\frac{1}{x^{2}}$ from $0$ .

$- \frac{1}{x^{2}} - 18 \frac{1}{x^{3}} - 3 \frac{1}{x^{4}}$

Step 1.1.5.4.2

Combine $- 18$ and $\frac{1}{x^{3}}$ .

$- \frac{1}{x^{2}} + \frac{- 18}{x^{3}} - 3 \frac{1}{x^{4}}$

Step 1.1.5.4.3

Move the negative in front of the fraction.

$- \frac{1}{x^{2}} - \frac{18}{x^{3}} - 3 \frac{1}{x^{4}}$

Step 1.1.5.4.4

Combine $- 3$ and $\frac{1}{x^{4}}$ .

$- \frac{1}{x^{2}} - \frac{18}{x^{3}} + \frac{- 3}{x^{4}}$

Step 1.1.5.4.5

Move the negative in front of the fraction.

$f' (x) = - \frac{1}{x^{2}} - \frac{18}{x^{3}} - \frac{3}{x^{4}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{1}{x^{2}} - \frac{18}{x^{3}} - \frac{3}{x^{4}}$ .

$- \frac{1}{x^{2}} - \frac{18}{x^{3}} - \frac{3}{x^{4}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{1}{x^{2}} - \frac{18}{x^{3}} - \frac{3}{x^{4}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{1}{x^{2}} - \frac{18}{x^{3}} - \frac{3}{x^{4}} = 0$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$x^{2}, x^{3}, x^{4}, 1$

Step 2.2.2

Since $x^{2}, x^{3}, x^{4}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1, 1$ then find LCM for the variable part $x^{2}, x^{3}, x^{4}$ .

Step 2.2.3

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 2.2.4

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

Not prime

Step 2.2.5

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

$1$

Step 2.2.6

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

Step 2.2.7

The factors for $x^{3}$ are $x \cdot x \cdot x$ , which is $x$ multiplied by each other $3$ times.

$x^{3} = x \cdot x \cdot x$

$x$ occurs $3$ times.

Step 2.2.8

The factors for $x^{4}$ are $x \cdot x \cdot x \cdot x$ , which is $x$ multiplied by each other $4$ times.

$x^{4} = x \cdot x \cdot x \cdot x$

$x$ occurs $4$ times.

Step 2.2.9

The LCM of $x^{2}, x^{3}, x^{4}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x \cdot x \cdot x$

Step 2.2.10

Simplify $x \cdot x \cdot x \cdot x$ .

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Step 2.2.10.1

Multiply $x$ by $x$ .

$x^{2} \cdot x \cdot x$

Step 2.2.10.2

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

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Step 2.2.10.2.1

Multiply $x^{2}$ by $x$ .

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Step 2.2.10.2.1.1

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

$x^{2} \cdot x^{1} \cdot x$

Step 2.2.10.2.1.2

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

$x^{2 + 1} \cdot x$

Step 2.2.10.2.2

Add $2$ and $1$ .

$x^{3} \cdot x$

Step 2.2.10.3

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 2.2.10.3.1

Multiply $x^{3}$ by $x$ .

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Step 2.2.10.3.1.1

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

$x^{3} \cdot x^{1}$

Step 2.2.10.3.1.2

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

$x^{3 + 1}$

Step 2.2.10.3.2

Add $3$ and $1$ .

$x^{4}$

Step 2.3

Multiply each term in $- \frac{1}{x^{2}} - \frac{18}{x^{3}} - \frac{3}{x^{4}} = 0$ by $x^{4}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $- \frac{1}{x^{2}} - \frac{18}{x^{3}} - \frac{3}{x^{4}} = 0$ by $x^{4}$ .

$- \frac{1}{x^{2}} x^{4} - \frac{18}{x^{3}} x^{4} - \frac{3}{x^{4}} x^{4} = 0 x^{4}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Cancel the common factor of $x^{2}$ .

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Step 2.3.2.1.1.1

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

$\frac{- 1}{x^{2}} x^{4} - \frac{18}{x^{3}} x^{4} - \frac{3}{x^{4}} x^{4} = 0 x^{4}$

Step 2.3.2.1.1.2

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

$\frac{- 1}{x^{2}} (x^{2} x^{2}) - \frac{18}{x^{3}} x^{4} - \frac{3}{x^{4}} x^{4} = 0 x^{4}$

Step 2.3.2.1.1.3

Cancel the common factor.

$\frac{- 1}{x^{2}} (x^{2} x^{2}) - \frac{18}{x^{3}} x^{4} - \frac{3}{x^{4}} x^{4} = 0 x^{4}$

Step 2.3.2.1.1.4

Rewrite the expression.

$- x^{2} - \frac{18}{x^{3}} x^{4} - \frac{3}{x^{4}} x^{4} = 0 x^{4}$

Step 2.3.2.1.2

Cancel the common factor of $x^{3}$ .

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Step 2.3.2.1.2.1

Move the leading negative in $- \frac{18}{x^{3}}$ into the numerator.

$- x^{2} + \frac{- 18}{x^{3}} x^{4} - \frac{3}{x^{4}} x^{4} = 0 x^{4}$

Step 2.3.2.1.2.2

Factor $x^{3}$ out of $x^{4}$ .

$- x^{2} + \frac{- 18}{x^{3}} (x^{3} x) - \frac{3}{x^{4}} x^{4} = 0 x^{4}$

Step 2.3.2.1.2.3

Cancel the common factor.

$- x^{2} + \frac{- 18}{x^{3}} (x^{3} x) - \frac{3}{x^{4}} x^{4} = 0 x^{4}$

Step 2.3.2.1.2.4

Rewrite the expression.

$- x^{2} - 18 x - \frac{3}{x^{4}} x^{4} = 0 x^{4}$

Step 2.3.2.1.3

Cancel the common factor of $x^{4}$ .

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Step 2.3.2.1.3.1

Move the leading negative in $- \frac{3}{x^{4}}$ into the numerator.

$- x^{2} - 18 x + \frac{- 3}{x^{4}} x^{4} = 0 x^{4}$

Step 2.3.2.1.3.2

Cancel the common factor.

$- x^{2} - 18 x + \frac{- 3}{x^{4}} x^{4} = 0 x^{4}$

Step 2.3.2.1.3.3

Rewrite the expression.

$- x^{2} - 18 x - 3 = 0 x^{4}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0$ by $x^{4}$ .

$- x^{2} - 18 x - 3 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.4.2

Substitute the values $a = - 1$ , $b = - 18$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{18 \pm \sqrt{{(- 18)}^{2} - 4 \cdot (- 1 \cdot - 3)}}{2 \cdot - 1}$

Step 2.4.3

Simplify.

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Step 2.4.3.1

Simplify the numerator.

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Step 2.4.3.1.1

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot - 1 \cdot - 3}}{2 \cdot - 1}$

Step 2.4.3.1.2

Multiply $- 4 \cdot - 1 \cdot - 3$ .

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Step 2.4.3.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{18 \pm \sqrt{324 + 4 \cdot - 3}}{2 \cdot - 1}$

Step 2.4.3.1.2.2

Multiply $4$ by $- 3$ .

$x = \frac{18 \pm \sqrt{324 - 12}}{2 \cdot - 1}$

Step 2.4.3.1.3

Subtract $12$ from $324$ .

$x = \frac{18 \pm \sqrt{312}}{2 \cdot - 1}$

Step 2.4.3.1.4

Rewrite $312$ as $2^{2} \cdot 78$ .

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Step 2.4.3.1.4.1

Factor $4$ out of $312$ .

$x = \frac{18 \pm \sqrt{4 (78)}}{2 \cdot - 1}$

Step 2.4.3.1.4.2

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

$x = \frac{18 \pm \sqrt{2^{2} \cdot 78}}{2 \cdot - 1}$

Step 2.4.3.1.5

Pull terms out from under the radical.

$x = \frac{18 \pm 2 \sqrt{78}}{2 \cdot - 1}$

Step 2.4.3.2

Multiply $2$ by $- 1$ .

$x = \frac{18 \pm 2 \sqrt{78}}{- 2}$

Step 2.4.3.3

Simplify $\frac{18 \pm 2 \sqrt{78}}{- 2}$ .

$x = \frac{9 \pm \sqrt{78}}{- 1}$

Step 2.4.3.4

Move the negative one from the denominator of $\frac{9 \pm \sqrt{78}}{- 1}$ .

$x = - 1 \cdot (9 \pm \sqrt{78})$

Step 2.4.3.5

Rewrite $- 1 \cdot (9 \pm \sqrt{78})$ as $- (9 \pm \sqrt{78})$ .

$x = - (9 \pm \sqrt{78})$

Step 2.4.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.4.4.1

Simplify the numerator.

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Step 2.4.4.1.1

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot - 1 \cdot - 3}}{2 \cdot - 1}$

Step 2.4.4.1.2

Multiply $- 4 \cdot - 1 \cdot - 3$ .

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Step 2.4.4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{18 \pm \sqrt{324 + 4 \cdot - 3}}{2 \cdot - 1}$

Step 2.4.4.1.2.2

Multiply $4$ by $- 3$ .

$x = \frac{18 \pm \sqrt{324 - 12}}{2 \cdot - 1}$

Step 2.4.4.1.3

Subtract $12$ from $324$ .

$x = \frac{18 \pm \sqrt{312}}{2 \cdot - 1}$

Step 2.4.4.1.4

Rewrite $312$ as $2^{2} \cdot 78$ .

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Step 2.4.4.1.4.1

Factor $4$ out of $312$ .

$x = \frac{18 \pm \sqrt{4 (78)}}{2 \cdot - 1}$

Step 2.4.4.1.4.2

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

$x = \frac{18 \pm \sqrt{2^{2} \cdot 78}}{2 \cdot - 1}$

Step 2.4.4.1.5

Pull terms out from under the radical.

$x = \frac{18 \pm 2 \sqrt{78}}{2 \cdot - 1}$

Step 2.4.4.2

Multiply $2$ by $- 1$ .

$x = \frac{18 \pm 2 \sqrt{78}}{- 2}$

Step 2.4.4.3

Simplify $\frac{18 \pm 2 \sqrt{78}}{- 2}$ .

$x = \frac{9 \pm \sqrt{78}}{- 1}$

Step 2.4.4.4

Move the negative one from the denominator of $\frac{9 \pm \sqrt{78}}{- 1}$ .

$x = - 1 \cdot (9 \pm \sqrt{78})$

Step 2.4.4.5

Rewrite $- 1 \cdot (9 \pm \sqrt{78})$ as $- (9 \pm \sqrt{78})$ .

$x = - (9 \pm \sqrt{78})$

Step 2.4.4.6

Change the $\pm$ to $+$ .

$x = - (9 + \sqrt{78})$

Step 2.4.4.7

Apply the distributive property.

$x = - 1 \cdot 9 - \sqrt{78}$

Step 2.4.4.8

Multiply $- 1$ by $9$ .

$x = - 9 - \sqrt{78}$

Step 2.4.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.4.5.1

Simplify the numerator.

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Step 2.4.5.1.1

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot - 1 \cdot - 3}}{2 \cdot - 1}$

Step 2.4.5.1.2

Multiply $- 4 \cdot - 1 \cdot - 3$ .

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Step 2.4.5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{18 \pm \sqrt{324 + 4 \cdot - 3}}{2 \cdot - 1}$

Step 2.4.5.1.2.2

Multiply $4$ by $- 3$ .

$x = \frac{18 \pm \sqrt{324 - 12}}{2 \cdot - 1}$

Step 2.4.5.1.3

Subtract $12$ from $324$ .

$x = \frac{18 \pm \sqrt{312}}{2 \cdot - 1}$

Step 2.4.5.1.4

Rewrite $312$ as $2^{2} \cdot 78$ .

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Step 2.4.5.1.4.1

Factor $4$ out of $312$ .

$x = \frac{18 \pm \sqrt{4 (78)}}{2 \cdot - 1}$

Step 2.4.5.1.4.2

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

$x = \frac{18 \pm \sqrt{2^{2} \cdot 78}}{2 \cdot - 1}$

Step 2.4.5.1.5

Pull terms out from under the radical.

$x = \frac{18 \pm 2 \sqrt{78}}{2 \cdot - 1}$

Step 2.4.5.2

Multiply $2$ by $- 1$ .

$x = \frac{18 \pm 2 \sqrt{78}}{- 2}$

Step 2.4.5.3

Simplify $\frac{18 \pm 2 \sqrt{78}}{- 2}$ .

$x = \frac{9 \pm \sqrt{78}}{- 1}$

Step 2.4.5.4

Move the negative one from the denominator of $\frac{9 \pm \sqrt{78}}{- 1}$ .

$x = - 1 \cdot (9 \pm \sqrt{78})$

Step 2.4.5.5

Rewrite $- 1 \cdot (9 \pm \sqrt{78})$ as $- (9 \pm \sqrt{78})$ .

$x = - (9 \pm \sqrt{78})$

Step 2.4.5.6

Change the $\pm$ to $-$ .

$x = - (9 - \sqrt{78})$

Step 2.4.5.7

Apply the distributive property.

$x = - 1 \cdot 9 + \sqrt{78}$

Step 2.4.5.8

Multiply $- 1$ by $9$ .

$x = - 9 + \sqrt{78}$

Step 2.4.5.9

Multiply $- - \sqrt{78}$ .

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Step 2.4.5.9.1

Multiply $- 1$ by $- 1$ .

$x = - 9 + 1 \sqrt{78}$

Step 2.4.5.9.2

Multiply $\sqrt{78}$ by $1$ .

$x = - 9 + \sqrt{78}$

Step 2.4.6

The final answer is the combination of both solutions.

$x = - 9 - \sqrt{78}, - 9 + \sqrt{78}$

Step 3

The values which make the derivative equal to $0$ are $- 9 - \sqrt{78}, - 9 + \sqrt{78}$ .

$- 9 - \sqrt{78}, - 9 + \sqrt{78}$

Step 4

Find where the derivative is undefined.

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Step 4.1

Set the denominator in $\frac{1}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 4.2

Solve for $x$ .

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Step 4.2.1

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

$x = \pm \sqrt{0}$

Step 4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

$x = \pm 0$

Step 4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4.3

Set the denominator in $\frac{18}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

Step 4.4

Solve for $x$ .

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Step 4.4.1

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

$x = \sqrt[3]{0}$

Step 4.4.2

Simplify $\sqrt[3]{0}$ .

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Step 4.4.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 4.4.2.2

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

$x = 0$

Step 4.5

Set the denominator in $\frac{3}{x^{4}}$ equal to $0$ to find where the expression is undefined.

$x^{4} = 0$

Step 4.6

Solve for $x$ .

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Step 4.6.1

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

$x = \pm \sqrt[4]{0}$

Step 4.6.2

Simplify $\pm \sqrt[4]{0}$ .

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Step 4.6.2.1

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 4.6.2.2

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

$x = \pm 0$

Step 4.6.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

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

$(- \infty, - 9 - \sqrt{78}) \cup (- 9 - \sqrt{78}, - 9 + \sqrt{78}) \cup (- 9 + \sqrt{78}, 0) \cup (0, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 9 - \sqrt{78})$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 18.83176$ in the expression.

$f' (- 18.83176) = - \frac{1}{{(- 18.83176)}^{2}} - \frac{18}{{(- 18.83176)}^{3}} - \frac{3}{{(- 18.83176)}^{4}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Find the common denominator.

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Step 6.2.1.1

Multiply $\frac{1}{{(- 18.83176)}^{2}}$ by $\frac{{(- 18.83176)}^{2}}{{(- 18.83176)}^{2}}$ .

$f' (- 18.83176) = - (\frac{1}{{(- 18.83176)}^{2}} \cdot \frac{{(- 18.83176)}^{2}}{{(- 18.83176)}^{2}}) - \frac{18}{{(- 18.83176)}^{3}} - \frac{3}{{(- 18.83176)}^{4}}$

Step 6.2.1.2

Multiply $\frac{1}{{(- 18.83176)}^{2}}$ by $\frac{{(- 18.83176)}^{2}}{{(- 18.83176)}^{2}}$ .

$f' (- 18.83176) = - \frac{{(- 18.83176)}^{2}}{{(- 18.83176)}^{2} {(- 18.83176)}^{2}} - \frac{18}{{(- 18.83176)}^{3}} - \frac{3}{{(- 18.83176)}^{4}}$

Step 6.2.1.3

Multiply $\frac{18}{{(- 18.83176)}^{3}}$ by $\frac{- 18.83176}{- 18.83176}$ .

$f' (- 18.83176) = - \frac{{(- 18.83176)}^{2}}{{(- 18.83176)}^{2} {(- 18.83176)}^{2}} - (\frac{18}{{(- 18.83176)}^{3}} \cdot \frac{- 18.83176}{- 18.83176}) - \frac{3}{{(- 18.83176)}^{4}}$

Step 6.2.1.4

Multiply $\frac{18}{{(- 18.83176)}^{3}}$ by $\frac{- 18.83176}{- 18.83176}$ .

$f' (- 18.83176) = - \frac{{(- 18.83176)}^{2}}{{(- 18.83176)}^{2} {(- 18.83176)}^{2}} - \frac{18 \cdot - 18.83176}{{(- 18.83176)}^{3} \cdot - 18.83176} - \frac{3}{{(- 18.83176)}^{4}}$

Step 6.2.1.5

Multiply ${(- 18.83176)}^{2}$ by ${(- 18.83176)}^{2}$ by adding the exponents.

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Step 6.2.1.5.1

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

$f' (- 18.83176) = - \frac{{(- 18.83176)}^{2}}{{(- 18.83176)}^{2 + 2}} - \frac{18 \cdot - 18.83176}{{(- 18.83176)}^{3} \cdot - 18.83176} - \frac{3}{{(- 18.83176)}^{4}}$

Step 6.2.1.5.2

Add $2$ and $2$ .

$f' (- 18.83176) = - \frac{{(- 18.83176)}^{2}}{{(- 18.83176)}^{4}} - \frac{18 \cdot - 18.83176}{{(- 18.83176)}^{3} \cdot - 18.83176} - \frac{3}{{(- 18.83176)}^{4}}$

Step 6.2.1.6

Reorder the factors of ${(- 18.83176)}^{3} \cdot - 18.83176$ .

$f' (- 18.83176) = - \frac{{(- 18.83176)}^{2}}{{(- 18.83176)}^{4}} - \frac{18 \cdot - 18.83176}{- 18.83176 {(- 18.83176)}^{3}} - \frac{3}{{(- 18.83176)}^{4}}$

Step 6.2.1.7

Multiply $- 18.83176$ by ${(- 18.83176)}^{3}$ by adding the exponents.

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Step 6.2.1.7.1

Multiply $- 18.83176$ by ${(- 18.83176)}^{3}$ .

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Step 6.2.1.7.1.1

Raise $- 18.83176$ to the power of $1$ .

$f' (- 18.83176) = - \frac{{(- 18.83176)}^{2}}{{(- 18.83176)}^{4}} - \frac{18 \cdot - 18.83176}{(- 18.83176) {(- 18.83176)}^{3}} - \frac{3}{{(- 18.83176)}^{4}}$

Step 6.2.1.7.1.2

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

$f' (- 18.83176) = - \frac{{(- 18.83176)}^{2}}{{(- 18.83176)}^{4}} - \frac{18 \cdot - 18.83176}{{(- 18.83176)}^{1 + 3}} - \frac{3}{{(- 18.83176)}^{4}}$

Step 6.2.1.7.2

Add $1$ and $3$ .

$f' (- 18.83176) = - \frac{{(- 18.83176)}^{2}}{{(- 18.83176)}^{4}} - \frac{18 \cdot - 18.83176}{{(- 18.83176)}^{4}} - \frac{3}{{(- 18.83176)}^{4}}$

Step 6.2.2

Combine the numerators over the common denominator.

$f' (- 18.83176) = \frac{- {(- 18.83176)}^{2} - 18 \cdot - 18.83176 - 3}{{(- 18.83176)}^{4}}$

Step 6.2.3

Simplify each term.

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Step 6.2.3.1

Raise $- 18.83176$ to the power of $2$ .

$f' (- 18.83176) = \frac{- 1 \cdot 354.63518469 - 18 \cdot - 18.83176 - 3}{{(- 18.83176)}^{4}}$

Step 6.2.3.2

Multiply $- 1$ by $354.63518469$ .

$f' (- 18.83176) = \frac{- 354.63518469 - 18 \cdot - 18.83176 - 3}{{(- 18.83176)}^{4}}$

Step 6.2.3.3

Multiply $- 18$ by $- 18.83176$ .

$f' (- 18.83176) = \frac{- 354.63518469 + 338.97168 - 3}{{(- 18.83176)}^{4}}$

Step 6.2.4

Simplify the expression.

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Step 6.2.4.1

Add $- 354.63518469$ and $338.97168$ .

$f' (- 18.83176) = \frac{- 15.66350469 - 3}{{(- 18.83176)}^{4}}$

Step 6.2.4.2

Subtract $3$ from $- 15.66350469$ .

$f' (- 18.83176) = \frac{- 18.66350469}{{(- 18.83176)}^{4}}$

Step 6.2.4.3

Raise $- 18.83176$ to the power of $4$ .

$f' (- 18.83176) = \frac{- 18.66350469}{125766.1142255}$

Step 6.2.4.4

Divide $- 18.66350469$ by $125766.1142255$ .

$f' (- 18.83176) = - 0.00014839$

Step 6.2.5

The final answer is $- 0.00014839$ .

$- 0.00014839$

Step 6.3

At $x = - 18.83176$ the derivative is $- 0.00014839$ . Since this is negative, the function is decreasing on $(- \infty, - 9 - \sqrt{78})$ .

Decreasing on $(- \infty, - 9 - \sqrt{78})$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(- 17.83176, - 9 + \sqrt{78})$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 8.99999956$ in the expression.

$f' (- 8.99999956) = - \frac{1}{{(- 8.99999956)}^{2}} - \frac{18}{{(- 8.99999956)}^{3}} - \frac{3}{{(- 8.99999956)}^{4}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Find the common denominator.

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Step 7.2.1.1

Multiply $\frac{1}{{(- 8.99999956)}^{2}}$ by $\frac{{(- 8.99999956)}^{2}}{{(- 8.99999956)}^{2}}$ .

$f' (- 8.99999956) = - (\frac{1}{{(- 8.99999956)}^{2}} \cdot \frac{{(- 8.99999956)}^{2}}{{(- 8.99999956)}^{2}}) - \frac{18}{{(- 8.99999956)}^{3}} - \frac{3}{{(- 8.99999956)}^{4}}$

Step 7.2.1.2

Multiply $\frac{1}{{(- 8.99999956)}^{2}}$ by $\frac{{(- 8.99999956)}^{2}}{{(- 8.99999956)}^{2}}$ .

$f' (- 8.99999956) = - \frac{{(- 8.99999956)}^{2}}{{(- 8.99999956)}^{2} {(- 8.99999956)}^{2}} - \frac{18}{{(- 8.99999956)}^{3}} - \frac{3}{{(- 8.99999956)}^{4}}$

Step 7.2.1.3

Multiply $\frac{18}{{(- 8.99999956)}^{3}}$ by $\frac{- 8.99999956}{- 8.99999956}$ .

$f' (- 8.99999956) = - \frac{{(- 8.99999956)}^{2}}{{(- 8.99999956)}^{2} {(- 8.99999956)}^{2}} - (\frac{18}{{(- 8.99999956)}^{3}} \cdot \frac{- 8.99999956}{- 8.99999956}) - \frac{3}{{(- 8.99999956)}^{4}}$

Step 7.2.1.4

Multiply $\frac{18}{{(- 8.99999956)}^{3}}$ by $\frac{- 8.99999956}{- 8.99999956}$ .

$f' (- 8.99999956) = - \frac{{(- 8.99999956)}^{2}}{{(- 8.99999956)}^{2} {(- 8.99999956)}^{2}} - \frac{18 \cdot - 8.99999956}{{(- 8.99999956)}^{3} \cdot - 8.99999956} - \frac{3}{{(- 8.99999956)}^{4}}$

Step 7.2.1.5

Multiply ${(- 8.99999956)}^{2}$ by ${(- 8.99999956)}^{2}$ by adding the exponents.

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Step 7.2.1.5.1

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

$f' (- 8.99999956) = - \frac{{(- 8.99999956)}^{2}}{{(- 8.99999956)}^{2 + 2}} - \frac{18 \cdot - 8.99999956}{{(- 8.99999956)}^{3} \cdot - 8.99999956} - \frac{3}{{(- 8.99999956)}^{4}}$

Step 7.2.1.5.2

Add $2$ and $2$ .

$f' (- 8.99999956) = - \frac{{(- 8.99999956)}^{2}}{{(- 8.99999956)}^{4}} - \frac{18 \cdot - 8.99999956}{{(- 8.99999956)}^{3} \cdot - 8.99999956} - \frac{3}{{(- 8.99999956)}^{4}}$

Step 7.2.1.6

Reorder the factors of ${(- 8.99999956)}^{3} \cdot - 8.99999956$ .

$f' (- 8.99999956) = - \frac{{(- 8.99999956)}^{2}}{{(- 8.99999956)}^{4}} - \frac{18 \cdot - 8.99999956}{- 8.99999956 {(- 8.99999956)}^{3}} - \frac{3}{{(- 8.99999956)}^{4}}$

Step 7.2.1.7

Multiply $- 8.99999956$ by ${(- 8.99999956)}^{3}$ by adding the exponents.

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Step 7.2.1.7.1

Multiply $- 8.99999956$ by ${(- 8.99999956)}^{3}$ .

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Step 7.2.1.7.1.1

Raise $- 8.99999956$ to the power of $1$ .

$f' (- 8.99999956) = - \frac{{(- 8.99999956)}^{2}}{{(- 8.99999956)}^{4}} - \frac{18 \cdot - 8.99999956}{(- 8.99999956) {(- 8.99999956)}^{3}} - \frac{3}{{(- 8.99999956)}^{4}}$

Step 7.2.1.7.1.2

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

$f' (- 8.99999956) = - \frac{{(- 8.99999956)}^{2}}{{(- 8.99999956)}^{4}} - \frac{18 \cdot - 8.99999956}{{(- 8.99999956)}^{1 + 3}} - \frac{3}{{(- 8.99999956)}^{4}}$

Step 7.2.1.7.2

Add $1$ and $3$ .

$f' (- 8.99999956) = - \frac{{(- 8.99999956)}^{2}}{{(- 8.99999956)}^{4}} - \frac{18 \cdot - 8.99999956}{{(- 8.99999956)}^{4}} - \frac{3}{{(- 8.99999956)}^{4}}$

Step 7.2.2

Combine the numerators over the common denominator.

$f' (- 8.99999956) = \frac{- {(- 8.99999956)}^{2} - 18 \cdot - 8.99999956 - 3}{{(- 8.99999956)}^{4}}$

Step 7.2.3

Simplify each term.

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Step 7.2.3.1

Raise $- 8.99999956$ to the power of $2$ .

$f' (- 8.99999956) = \frac{- 1 \cdot 80.99999217 - 18 \cdot - 8.99999956 - 3}{{(- 8.99999956)}^{4}}$

Step 7.2.3.2

Multiply $- 1$ by $80.99999217$ .

$f' (- 8.99999956) = \frac{- 80.99999217 - 18 \cdot - 8.99999956 - 3}{{(- 8.99999956)}^{4}}$

Step 7.2.3.3

Multiply $- 18$ by $- 8.99999956$ .

$f' (- 8.99999956) = \frac{- 80.99999217 + 161.99999217 - 3}{{(- 8.99999956)}^{4}}$

Step 7.2.4

Simplify the expression.

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Step 7.2.4.1

Add $- 80.99999217$ and $161.99999217$ .

$f' (- 8.99999956) = \frac{81 - 3}{{(- 8.99999956)}^{4}}$

Step 7.2.4.2

Subtract $3$ from $81$ .

$f' (- 8.99999956) = \frac{78}{{(- 8.99999956)}^{4}}$

Step 7.2.4.3

Raise $- 8.99999956$ to the power of $4$ .

$f' (- 8.99999956) = \frac{78}{6560.99873154}$

Step 7.2.4.4

Divide $78$ by $6560.99873154$ .

$f' (- 8.99999956) = 0.01188843$

Step 7.2.5

The final answer is $0.01188843$ .

$0.01188843$

Step 7.3

At $x = - 8.99999956$ the derivative is $0.01188843$ . Since this is positive, the function is increasing on $(- 17.83176, - 9 + \sqrt{78})$ .

Increasing on $(- 9 - \sqrt{78}, - 9 + \sqrt{78})$ since $f' (x) > 0$

Step 8

Substitute a value from the interval $(- 0.16823913, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $- 0.08411956$ in the expression.

$f' (- 0.08411956) = - \frac{1}{{(- 0.08411956)}^{2}} - \frac{18}{{(- 0.08411956)}^{3}} - \frac{3}{{(- 0.08411956)}^{4}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Find the common denominator.

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Step 8.2.1.1

Multiply $\frac{1}{{(- 0.08411956)}^{2}}$ by $\frac{{(- 0.08411956)}^{2}}{{(- 0.08411956)}^{2}}$ .

$f' (- 0.08411956) = - (\frac{1}{{(- 0.08411956)}^{2}} \cdot \frac{{(- 0.08411956)}^{2}}{{(- 0.08411956)}^{2}}) - \frac{18}{{(- 0.08411956)}^{3}} - \frac{3}{{(- 0.08411956)}^{4}}$

Step 8.2.1.2

Multiply $\frac{1}{{(- 0.08411956)}^{2}}$ by $\frac{{(- 0.08411956)}^{2}}{{(- 0.08411956)}^{2}}$ .

$f' (- 0.08411956) = - \frac{{(- 0.08411956)}^{2}}{{(- 0.08411956)}^{2} {(- 0.08411956)}^{2}} - \frac{18}{{(- 0.08411956)}^{3}} - \frac{3}{{(- 0.08411956)}^{4}}$

Step 8.2.1.3

Multiply $\frac{18}{{(- 0.08411956)}^{3}}$ by $\frac{- 0.08411956}{- 0.08411956}$ .

$f' (- 0.08411956) = - \frac{{(- 0.08411956)}^{2}}{{(- 0.08411956)}^{2} {(- 0.08411956)}^{2}} - (\frac{18}{{(- 0.08411956)}^{3}} \cdot \frac{- 0.08411956}{- 0.08411956}) - \frac{3}{{(- 0.08411956)}^{4}}$

Step 8.2.1.4

Multiply $\frac{18}{{(- 0.08411956)}^{3}}$ by $\frac{- 0.08411956}{- 0.08411956}$ .

$f' (- 0.08411956) = - \frac{{(- 0.08411956)}^{2}}{{(- 0.08411956)}^{2} {(- 0.08411956)}^{2}} - \frac{18 \cdot - 0.08411956}{{(- 0.08411956)}^{3} \cdot - 0.08411956} - \frac{3}{{(- 0.08411956)}^{4}}$

Step 8.2.1.5

Multiply ${(- 0.08411956)}^{2}$ by ${(- 0.08411956)}^{2}$ by adding the exponents.

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Step 8.2.1.5.1

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

$f' (- 0.08411956) = - \frac{{(- 0.08411956)}^{2}}{{(- 0.08411956)}^{2 + 2}} - \frac{18 \cdot - 0.08411956}{{(- 0.08411956)}^{3} \cdot - 0.08411956} - \frac{3}{{(- 0.08411956)}^{4}}$

Step 8.2.1.5.2

Add $2$ and $2$ .

$f' (- 0.08411956) = - \frac{{(- 0.08411956)}^{2}}{{(- 0.08411956)}^{4}} - \frac{18 \cdot - 0.08411956}{{(- 0.08411956)}^{3} \cdot - 0.08411956} - \frac{3}{{(- 0.08411956)}^{4}}$

Step 8.2.1.6

Reorder the factors of ${(- 0.08411956)}^{3} \cdot - 0.08411956$ .

$f' (- 0.08411956) = - \frac{{(- 0.08411956)}^{2}}{{(- 0.08411956)}^{4}} - \frac{18 \cdot - 0.08411956}{- 0.08411956 {(- 0.08411956)}^{3}} - \frac{3}{{(- 0.08411956)}^{4}}$

Step 8.2.1.7

Multiply $- 0.08411956$ by ${(- 0.08411956)}^{3}$ by adding the exponents.

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Step 8.2.1.7.1

Multiply $- 0.08411956$ by ${(- 0.08411956)}^{3}$ .

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Step 8.2.1.7.1.1

Raise $- 0.08411956$ to the power of $1$ .

$f' (- 0.08411956) = - \frac{{(- 0.08411956)}^{2}}{{(- 0.08411956)}^{4}} - \frac{18 \cdot - 0.08411956}{(- 0.08411956) {(- 0.08411956)}^{3}} - \frac{3}{{(- 0.08411956)}^{4}}$

Step 8.2.1.7.1.2

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

$f' (- 0.08411956) = - \frac{{(- 0.08411956)}^{2}}{{(- 0.08411956)}^{4}} - \frac{18 \cdot - 0.08411956}{{(- 0.08411956)}^{1 + 3}} - \frac{3}{{(- 0.08411956)}^{4}}$

Step 8.2.1.7.2

Add $1$ and $3$ .

$f' (- 0.08411956) = - \frac{{(- 0.08411956)}^{2}}{{(- 0.08411956)}^{4}} - \frac{18 \cdot - 0.08411956}{{(- 0.08411956)}^{4}} - \frac{3}{{(- 0.08411956)}^{4}}$

Step 8.2.2

Combine the numerators over the common denominator.

$f' (- 0.08411956) = \frac{- {(- 0.08411956)}^{2} - 18 \cdot - 0.08411956 - 3}{{(- 0.08411956)}^{4}}$

Step 8.2.3

Simplify each term.

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Step 8.2.3.1

Raise $- 0.08411956$ to the power of $2$ .

$f' (- 0.08411956) = \frac{- 1 \cdot 0.0070761 - 18 \cdot - 0.08411956 - 3}{{(- 0.08411956)}^{4}}$

Step 8.2.3.2

Multiply $- 1$ by $0.0070761$ .

$f' (- 0.08411956) = \frac{- 0.0070761 - 18 \cdot - 0.08411956 - 3}{{(- 0.08411956)}^{4}}$

Step 8.2.3.3

Multiply $- 18$ by $- 0.08411956$ .

$f' (- 0.08411956) = \frac{- 0.0070761 + 1.51415217 - 3}{{(- 0.08411956)}^{4}}$

Step 8.2.4

Simplify the expression.

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Step 8.2.4.1

Add $- 0.0070761$ and $1.51415217$ .

$f' (- 0.08411956) = \frac{1.50707606 - 3}{{(- 0.08411956)}^{4}}$

Step 8.2.4.2

Subtract $3$ from $1.50707606$ .

$f' (- 0.08411956) = \frac{- 1.49292393}{{(- 0.08411956)}^{4}}$

Step 8.2.4.3

Raise $- 0.08411956$ to the power of $4$ .

$f' (- 0.08411956) = \frac{- 1.49292393}{0.00005007}$

Step 8.2.4.4

Divide $- 1.49292393$ by $0.00005007$ .

$f' (- 0.08411956) = - 29816.01559941$

Step 8.2.5

The final answer is $- 29816.01559941$ .

$- 29816.01559941$

Step 8.3

At $x = - 0.08411956$ the derivative is $- 29816.01559941$ . Since this is negative, the function is decreasing on $(- 0.16823913, 0)$ .

Decreasing on $(- 9 + \sqrt{78}, 0)$ since $f' (x) < 0$

Step 9

Substitute a value from the interval $(0, \infty)$ 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.

$f' (1) = - \frac{1}{{(1)}^{2}} - \frac{18}{{(1)}^{3}} - \frac{3}{{(1)}^{4}}$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

One to any power is one.

$f' (1) = - \frac{1}{1} - \frac{18}{{(1)}^{3}} - \frac{3}{{(1)}^{4}}$

Step 9.2.1.2

Cancel the common factor of $1$ .

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Step 9.2.1.2.1

Cancel the common factor.

$f' (1) = - \frac{1}{1} - \frac{18}{{(1)}^{3}} - \frac{3}{{(1)}^{4}}$

Step 9.2.1.2.2

Rewrite the expression.

$f' (1) = - 1 \cdot 1 - \frac{18}{{(1)}^{3}} - \frac{3}{{(1)}^{4}}$

Step 9.2.1.3

Multiply $- 1$ by $1$ .

$f' (1) = - 1 - \frac{18}{{(1)}^{3}} - \frac{3}{{(1)}^{4}}$

Step 9.2.1.4

One to any power is one.

$f' (1) = - 1 - \frac{18}{1} - \frac{3}{{(1)}^{4}}$

Step 9.2.1.5

Divide $18$ by $1$ .

$f' (1) = - 1 - 1 \cdot 18 - \frac{3}{{(1)}^{4}}$

Step 9.2.1.6

Multiply $- 1$ by $18$ .

$f' (1) = - 1 - 18 - \frac{3}{{(1)}^{4}}$

Step 9.2.1.7

One to any power is one.

$f' (1) = - 1 - 18 - \frac{3}{1}$

Step 9.2.1.8

Divide $3$ by $1$ .

$f' (1) = - 1 - 18 - 1 \cdot 3$

Step 9.2.1.9

Multiply $- 1$ by $3$ .

$f' (1) = - 1 - 18 - 3$

Step 9.2.2

Simplify by subtracting numbers.

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Step 9.2.2.1

Subtract $18$ from $- 1$ .

$f' (1) = - 19 - 3$

Step 9.2.2.2

Subtract $3$ from $- 19$ .

$f' (1) = - 22$

Step 9.2.3

The final answer is $- 22$ .

$- 22$

Step 9.3

At $x = 1$ the derivative is $- 22$ . Since this is negative, the function is decreasing on $(0, \infty)$ .

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

Step 10

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

Increasing on: $(- 9 - \sqrt{78}, - 9 + \sqrt{78})$

Decreasing on: $(- \infty, - 9 - \sqrt{78}), (- 9 + \sqrt{78}, 0), (0, \infty)$

Step 11