Calculus Examples

$f (x) = 6 x^{- 3} (x^{4} - 6 x^{3} + 11 x - 2)$

Step 1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{- 3} (x^{4} - 6 x^{3} + 11 x - 2)$ with respect to $x$ is $6 \frac{d}{d x} [x^{- 3} (x^{4} - 6 x^{3} + 11 x - 2)]$ .

$6 \frac{d}{d x} [x^{- 3} (x^{4} - 6 x^{3} + 11 x - 2)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{- 3}$ and $g (x) = x^{4} - 6 x^{3} + 11 x - 2$ .

$6 (x^{- 3} \frac{d}{d x} [x^{4} - 6 x^{3} + 11 x - 2] + (x^{4} - 6 x^{3} + 11 x - 2) \frac{d}{d x} [x^{- 3}])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{4} - 6 x^{3} + 11 x - 2$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 6 x^{3}] + \frac{d}{d x} [11 x] + \frac{d}{d x} [- 2]$ .

$6 (x^{- 3} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 6 x^{3}] + \frac{d}{d x} [11 x] + \frac{d}{d x} [- 2]) + (x^{4} - 6 x^{3} + 11 x - 2) \frac{d}{d x} [x^{- 3}])$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$6 (x^{- 3} (4 x^{3} + \frac{d}{d x} [- 6 x^{3}] + \frac{d}{d x} [11 x] + \frac{d}{d x} [- 2]) + (x^{4} - 6 x^{3} + 11 x - 2) \frac{d}{d x} [x^{- 3}])$

Step 3.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{3}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{3}]$ .

$6 (x^{- 3} (4 x^{3} - 6 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [11 x] + \frac{d}{d x} [- 2]) + (x^{4} - 6 x^{3} + 11 x - 2) \frac{d}{d x} [x^{- 3}])$

Step 3.4

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

$6 (x^{- 3} (4 x^{3} - 6 (3 x^{2}) + \frac{d}{d x} [11 x] + \frac{d}{d x} [- 2]) + (x^{4} - 6 x^{3} + 11 x - 2) \frac{d}{d x} [x^{- 3}])$

Step 3.5

Multiply $3$ by $- 6$ .

$6 (x^{- 3} (4 x^{3} - 18 x^{2} + \frac{d}{d x} [11 x] + \frac{d}{d x} [- 2]) + (x^{4} - 6 x^{3} + 11 x - 2) \frac{d}{d x} [x^{- 3}])$

Step 3.6

Since $11$ is constant with respect to $x$ , the derivative of $11 x$ with respect to $x$ is $11 \frac{d}{d x} [x]$ .

$6 (x^{- 3} (4 x^{3} - 18 x^{2} + 11 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]) + (x^{4} - 6 x^{3} + 11 x - 2) \frac{d}{d x} [x^{- 3}])$

Step 3.7

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

$6 (x^{- 3} (4 x^{3} - 18 x^{2} + 11 \cdot 1 + \frac{d}{d x} [- 2]) + (x^{4} - 6 x^{3} + 11 x - 2) \frac{d}{d x} [x^{- 3}])$

Step 3.8

Multiply $11$ by $1$ .

$6 (x^{- 3} (4 x^{3} - 18 x^{2} + 11 + \frac{d}{d x} [- 2]) + (x^{4} - 6 x^{3} + 11 x - 2) \frac{d}{d x} [x^{- 3}])$

Step 3.9

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$6 (x^{- 3} (4 x^{3} - 18 x^{2} + 11 + 0) + (x^{4} - 6 x^{3} + 11 x - 2) \frac{d}{d x} [x^{- 3}])$

Step 3.10

Add $4 x^{3} - 18 x^{2} + 11$ and $0$ .

$6 (x^{- 3} (4 x^{3} - 18 x^{2} + 11) + (x^{4} - 6 x^{3} + 11 x - 2) \frac{d}{d x} [x^{- 3}])$

Step 3.11

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

$6 (x^{- 3} (4 x^{3} - 18 x^{2} + 11) + (x^{4} - 6 x^{3} + 11 x - 2) (- 3 x^{- 4}))$

Step 4

Simplify.

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Step 4.1

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

$6 (\frac{1}{x^{3}} (4 x^{3} - 18 x^{2} + 11) + (x^{4} - 6 x^{3} + 11 x - 2) (- 3 x^{- 4}))$

Step 4.2

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

$6 (\frac{1}{x^{3}} (4 x^{3} - 18 x^{2} + 11) + (x^{4} - 6 x^{3} + 11 x - 2) (- 3 \frac{1}{x^{4}}))$

Step 4.3

Apply the distributive property.

$6 (\frac{1}{x^{3}} (4 x^{3}) + \frac{1}{x^{3}} (- 18 x^{2}) + \frac{1}{x^{3}} \cdot 11 + (x^{4} - 6 x^{3} + 11 x - 2) (- 3 \frac{1}{x^{4}}))$

Step 4.4

Apply the distributive property.

$6 (\frac{1}{x^{3}} (4 x^{3}) + \frac{1}{x^{3}} (- 18 x^{2}) + \frac{1}{x^{3}} \cdot 11 + x^{4} (- 3 \frac{1}{x^{4}}) - 6 x^{3} (- 3 \frac{1}{x^{4}}) + 11 x (- 3 \frac{1}{x^{4}}) - 2 (- 3 \frac{1}{x^{4}}))$

Step 4.5

Apply the distributive property.

$6 (\frac{1}{x^{3}} (4 x^{3})) + 6 (\frac{1}{x^{3}} (- 18 x^{2})) + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6

Combine terms.

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Step 4.6.1

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

$6 (\frac{4}{x^{3}} x^{3}) + 6 (\frac{1}{x^{3}} (- 18 x^{2})) + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.2

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

$6 \frac{4 x^{3}}{x^{3}} + 6 (\frac{1}{x^{3}} (- 18 x^{2})) + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.3

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

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Step 4.6.3.1

Cancel the common factor.

Step 4.6.3.2

Divide $4$ by $1$ .

$6 \cdot 4 + 6 (\frac{1}{x^{3}} (- 18 x^{2})) + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.4

Multiply $6$ by $4$ .

$24 + 6 (\frac{1}{x^{3}} (- 18 x^{2})) + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.5

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

$24 + 6 (\frac{- 18}{x^{3}} x^{2}) + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.6

Combine $\frac{- 18}{x^{3}}$ and $x^{2}$ .

$24 + 6 \frac{- 18 x^{2}}{x^{3}} + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.7

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 4.6.7.1

Factor $x^{2}$ out of $- 18 x^{2}$ .

$24 + 6 \frac{x^{2} \cdot - 18}{x^{3}} + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.7.2

Cancel the common factors.

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Step 4.6.7.2.1

Factor $x^{2}$ out of $x^{3}$ .

$24 + 6 \frac{x^{2} \cdot - 18}{x^{2} x} + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.7.2.2

Cancel the common factor.

Step 4.6.7.2.3

Rewrite the expression.

$24 + 6 \frac{- 18}{x} + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.8

Move the negative in front of the fraction.

$24 + 6 (- \frac{18}{x}) + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.9

Multiply $- 1$ by $6$ .

$24 - 6 \frac{18}{x} + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.10

Combine $- 6$ and $\frac{18}{x}$ .

$24 + \frac{- 6 \cdot 18}{x} + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.11

Multiply $- 6$ by $18$ .

$24 + \frac{- 108}{x} + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.12

Move the negative in front of the fraction.

$24 - \frac{108}{x} + 6 (\frac{1}{x^{3}} \cdot 11) + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.13

Combine $\frac{1}{x^{3}}$ and $11$ .

$24 - \frac{108}{x} + 6 \frac{11}{x^{3}} + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.14

Combine $6$ and $\frac{11}{x^{3}}$ .

$24 - \frac{108}{x} + \frac{6 \cdot 11}{x^{3}} + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.15

Multiply $6$ by $11$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} + 6 (x^{4} (- 3 \frac{1}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.16

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

$24 - \frac{108}{x} + \frac{66}{x^{3}} + 6 (x^{4} \frac{- 3}{x^{4}}) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.17

Move the negative in front of the fraction.

$24 - \frac{108}{x} + \frac{66}{x^{3}} + 6 (x^{4} (- \frac{3}{x^{4}})) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.18

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

$24 - \frac{108}{x} + \frac{66}{x^{3}} + 6 (- \frac{x^{4} \cdot 3}{x^{4}}) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.19

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

$24 - \frac{108}{x} + \frac{66}{x^{3}} + 6 (- \frac{3 \cdot x^{4}}{x^{4}}) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.20

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

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Step 4.6.20.1

Cancel the common factor.

$24 - \frac{108}{x} + \frac{66}{x^{3}} + 6 (- \frac{3 x^{4}}{x^{4}}) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.20.2

Divide $3$ by $1$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} + 6 (- 1 \cdot 3) + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.21

Multiply $- 1$ by $3$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} + 6 \cdot - 3 + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.22

Multiply $6$ by $- 3$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + 6 (- 6 x^{3} (- 3 \frac{1}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.23

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

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + 6 (- 6 x^{3} \frac{- 3}{x^{4}}) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.24

Move the negative in front of the fraction.

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + 6 (- 6 x^{3} (- \frac{3}{x^{4}})) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.25

Multiply $- 1$ by $- 6$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + 6 (6 x^{3} \frac{3}{x^{4}}) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.26

Combine $6$ and $\frac{3}{x^{4}}$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + 6 (x^{3} \frac{6 \cdot 3}{x^{4}}) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.27

Multiply $6$ by $3$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + 6 (x^{3} \frac{18}{x^{4}}) + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.28

Combine $x^{3}$ and $\frac{18}{x^{4}}$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + 6 \frac{x^{3} \cdot 18}{x^{4}} + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.29

Move $18$ to the left of $x^{3}$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + 6 \frac{18 \cdot x^{3}}{x^{4}} + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.30

Cancel the common factor of $x^{3}$ and $x^{4}$ .

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Step 4.6.30.1

Factor $x^{3}$ out of $18 x^{3}$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + 6 \frac{x^{3} \cdot 18}{x^{4}} + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.30.2

Cancel the common factors.

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Step 4.6.30.2.1

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

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + 6 \frac{x^{3} \cdot 18}{x^{3} x} + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.30.2.2

Cancel the common factor.

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + 6 \frac{x^{3} \cdot 18}{x^{3} x} + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.30.2.3

Rewrite the expression.

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + 6 \frac{18}{x} + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.31

Combine $6$ and $\frac{18}{x}$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{6 \cdot 18}{x} + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.32

Multiply $6$ by $18$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + 6 (11 x (- 3 \frac{1}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.33

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

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + 6 (11 x \frac{- 3}{x^{4}}) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.34

Move the negative in front of the fraction.

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + 6 (11 x (- \frac{3}{x^{4}})) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.35

Multiply $- 1$ by $11$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + 6 (- 11 x \frac{3}{x^{4}}) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.36

Combine $- 11$ and $\frac{3}{x^{4}}$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + 6 (x \frac{- 11 \cdot 3}{x^{4}}) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.37

Multiply $- 11$ by $3$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + 6 (x \frac{- 33}{x^{4}}) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.38

Combine $x$ and $\frac{- 33}{x^{4}}$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + 6 \frac{x \cdot - 33}{x^{4}} + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.39

Move $- 33$ to the left of $x$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + 6 \frac{- 33 \cdot x}{x^{4}} + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.40

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

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Step 4.6.40.1

Factor $x$ out of $- 33 x$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + 6 \frac{x \cdot - 33}{x^{4}} + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.40.2

Cancel the common factors.

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Step 4.6.40.2.1

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

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + 6 \frac{x \cdot - 33}{x \cdot x^{3}} + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.40.2.2

Cancel the common factor.

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + 6 \frac{x \cdot - 33}{x \cdot x^{3}} + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.40.2.3

Rewrite the expression.

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + 6 \frac{- 33}{x^{3}} + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.41

Move the negative in front of the fraction.

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + 6 (- \frac{33}{x^{3}}) + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.42

Multiply $- 1$ by $6$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} - 6 \frac{33}{x^{3}} + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.43

Combine $- 6$ and $\frac{33}{x^{3}}$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + \frac{- 6 \cdot 33}{x^{3}} + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.44

Multiply $- 6$ by $33$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} + \frac{- 198}{x^{3}} + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.45

Move the negative in front of the fraction.

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} - \frac{198}{x^{3}} + 6 (- 2 (- 3 \frac{1}{x^{4}}))$

Step 4.6.46

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

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} - \frac{198}{x^{3}} + 6 (- 2 \frac{- 3}{x^{4}})$

Step 4.6.47

Move the negative in front of the fraction.

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} - \frac{198}{x^{3}} + 6 (- 2 (- \frac{3}{x^{4}}))$

Step 4.6.48

Multiply $- 1$ by $- 2$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} - \frac{198}{x^{3}} + 6 (2 \frac{3}{x^{4}})$

Step 4.6.49

Combine $2$ and $\frac{3}{x^{4}}$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} - \frac{198}{x^{3}} + 6 \frac{2 \cdot 3}{x^{4}}$

Step 4.6.50

Multiply $2$ by $3$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} - \frac{198}{x^{3}} + 6 \frac{6}{x^{4}}$

Step 4.6.51

Combine $6$ and $\frac{6}{x^{4}}$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} - \frac{198}{x^{3}} + \frac{6 \cdot 6}{x^{4}}$

Step 4.6.52

Multiply $6$ by $6$ .

$24 - \frac{108}{x} + \frac{66}{x^{3}} - 18 + \frac{108}{x} - \frac{198}{x^{3}} + \frac{36}{x^{4}}$

Step 4.6.53

Subtract $18$ from $24$ .

$- \frac{108}{x} + \frac{66}{x^{3}} + 6 + \frac{108}{x} - \frac{198}{x^{3}} + \frac{36}{x^{4}}$

Step 4.6.54

Add $- \frac{108}{x}$ and $\frac{108}{x}$ .

$\frac{66}{x^{3}} + 6 + 0 - \frac{198}{x^{3}} + \frac{36}{x^{4}}$

Step 4.6.55

Add $\frac{66}{x^{3}} + 6$ and $0$ .

$\frac{66}{x^{3}} + 6 - \frac{198}{x^{3}} + \frac{36}{x^{4}}$

Step 4.6.56

Combine the numerators over the common denominator.

$\frac{66 - 198}{x^{3}} + 6 + \frac{36}{x^{4}}$

Step 4.6.57

Subtract $198$ from $66$ .

$\frac{- 132}{x^{3}} + 6 + \frac{36}{x^{4}}$

Step 4.6.58

Move the negative in front of the fraction.

$- \frac{132}{x^{3}} + 6 + \frac{36}{x^{4}}$

Step 4.7

Reorder terms.

$- \frac{132}{x^{3}} + \frac{36}{x^{4}} + 6$