Calculus Examples

$f (x) = (10 - 7 x^{2}) (10 x^{3} + 6 x^{- 2} + 8)$

Step 1

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 $f (x) = 10 - 7 x^{2}$ and $g (x) = 10 x^{3} + 6 x^{- 2} + 8$ .

$(10 - 7 x^{2}) \frac{d}{d x} [10 x^{3} + 6 x^{- 2} + 8] + (10 x^{3} + 6 x^{- 2} + 8) \frac{d}{d x} [10 - 7 x^{2}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $10 x^{3} + 6 x^{- 2} + 8$ with respect to $x$ is $\frac{d}{d x} [10 x^{3}] + \frac{d}{d x} [6 x^{- 2}] + \frac{d}{d x} [8]$ .

$(10 - 7 x^{2}) (\frac{d}{d x} [10 x^{3}] + \frac{d}{d x} [6 x^{- 2}] + \frac{d}{d x} [8]) + (10 x^{3} + 6 x^{- 2} + 8) \frac{d}{d x} [10 - 7 x^{2}]$

Step 2.2

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{3}$ with respect to $x$ is $10 \frac{d}{d x} [x^{3}]$ .

$(10 - 7 x^{2}) (10 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [6 x^{- 2}] + \frac{d}{d x} [8]) + (10 x^{3} + 6 x^{- 2} + 8) \frac{d}{d x} [10 - 7 x^{2}]$

Step 2.3

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

$(10 - 7 x^{2}) (10 (3 x^{2}) + \frac{d}{d x} [6 x^{- 2}] + \frac{d}{d x} [8]) + (10 x^{3} + 6 x^{- 2} + 8) \frac{d}{d x} [10 - 7 x^{2}]$

Step 2.4

Multiply $3$ by $10$ .

$(10 - 7 x^{2}) (30 x^{2} + \frac{d}{d x} [6 x^{- 2}] + \frac{d}{d x} [8]) + (10 x^{3} + 6 x^{- 2} + 8) \frac{d}{d x} [10 - 7 x^{2}]$

Step 2.5

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

$(10 - 7 x^{2}) (30 x^{2} + 6 \frac{d}{d x} [x^{- 2}] + \frac{d}{d x} [8]) + (10 x^{3} + 6 x^{- 2} + 8) \frac{d}{d x} [10 - 7 x^{2}]$

Step 2.6

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

$(10 - 7 x^{2}) (30 x^{2} + 6 (- 2 x^{- 3}) + \frac{d}{d x} [8]) + (10 x^{3} + 6 x^{- 2} + 8) \frac{d}{d x} [10 - 7 x^{2}]$

Step 2.7

Multiply $- 2$ by $6$ .

$(10 - 7 x^{2}) (30 x^{2} - 12 x^{- 3} + \frac{d}{d x} [8]) + (10 x^{3} + 6 x^{- 2} + 8) \frac{d}{d x} [10 - 7 x^{2}]$

Step 2.8

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$(10 - 7 x^{2}) (30 x^{2} - 12 x^{- 3} + 0) + (10 x^{3} + 6 x^{- 2} + 8) \frac{d}{d x} [10 - 7 x^{2}]$

Step 2.9

Add $30 x^{2} - 12 x^{- 3}$ and $0$ .

$(10 - 7 x^{2}) (30 x^{2} - 12 x^{- 3}) + (10 x^{3} + 6 x^{- 2} + 8) \frac{d}{d x} [10 - 7 x^{2}]$

Step 2.10

By the Sum Rule, the derivative of $10 - 7 x^{2}$ with respect to $x$ is $\frac{d}{d x} [10] + \frac{d}{d x} [- 7 x^{2}]$ .

$(10 - 7 x^{2}) (30 x^{2} - 12 x^{- 3}) + (10 x^{3} + 6 x^{- 2} + 8) (\frac{d}{d x} [10] + \frac{d}{d x} [- 7 x^{2}])$

Step 2.11

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$(10 - 7 x^{2}) (30 x^{2} - 12 x^{- 3}) + (10 x^{3} + 6 x^{- 2} + 8) (0 + \frac{d}{d x} [- 7 x^{2}])$

Step 2.12

Add $0$ and $\frac{d}{d x} [- 7 x^{2}]$ .

$(10 - 7 x^{2}) (30 x^{2} - 12 x^{- 3}) + (10 x^{3} + 6 x^{- 2} + 8) \frac{d}{d x} [- 7 x^{2}]$

Step 2.13

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{2}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{2}]$ .

$(10 - 7 x^{2}) (30 x^{2} - 12 x^{- 3}) + (10 x^{3} + 6 x^{- 2} + 8) (- 7 \frac{d}{d x} [x^{2}])$

Step 2.14

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

$(10 - 7 x^{2}) (30 x^{2} - 12 x^{- 3}) + (10 x^{3} + 6 x^{- 2} + 8) (- 7 (2 x))$

Step 2.15

Multiply $2$ by $- 7$ .

$(10 - 7 x^{2}) (30 x^{2} - 12 x^{- 3}) + (10 x^{3} + 6 x^{- 2} + 8) (- 14 x)$

Step 3

Simplify.

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Step 3.1

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

$(10 - 7 x^{2}) (30 x^{2} - 12 \frac{1}{x^{3}}) + (10 x^{3} + 6 x^{- 2} + 8) (- 14 x)$

Step 3.2

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

$(10 - 7 x^{2}) (30 x^{2} - 12 \frac{1}{x^{3}}) + (10 x^{3} + 6 \frac{1}{x^{2}} + 8) (- 14 x)$

Step 3.3

Apply the distributive property.

$(10 - 7 x^{2}) (30 x^{2} - 12 \frac{1}{x^{3}}) + 10 x^{3} (- 14 x) + 6 \frac{1}{x^{2}} (- 14 x) + 8 (- 14 x)$

Step 3.4

Combine terms.

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Step 3.4.1

Combine $- 12$ and $\frac{1}{x^{3}}$ .

$(10 - 7 x^{2}) (30 x^{2} + \frac{- 12}{x^{3}}) + 10 x^{3} (- 14 x) + 6 \frac{1}{x^{2}} (- 14 x) + 8 (- 14 x)$

Step 3.4.2

Move the negative in front of the fraction.

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) + 10 x^{3} (- 14 x) + 6 \frac{1}{x^{2}} (- 14 x) + 8 (- 14 x)$

Step 3.4.3

Multiply $- 14$ by $10$ .

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 140 x^{3} x + 6 \frac{1}{x^{2}} (- 14 x) + 8 (- 14 x)$

Step 3.4.4

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

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 140 (x^{1} x^{3}) + 6 \frac{1}{x^{2}} (- 14 x) + 8 (- 14 x)$

Step 3.4.5

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

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 140 x^{1 + 3} + 6 \frac{1}{x^{2}} (- 14 x) + 8 (- 14 x)$

Step 3.4.6

Add $1$ and $3$ .

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 140 x^{4} + 6 \frac{1}{x^{2}} (- 14 x) + 8 (- 14 x)$

Step 3.4.7

Combine $6$ and $\frac{1}{x^{2}}$ .

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 140 x^{4} + \frac{6}{x^{2}} (- 14 x) + 8 (- 14 x)$

Step 3.4.8

Combine $- 14$ and $\frac{6}{x^{2}}$ .

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 140 x^{4} + \frac{- 14 \cdot 6}{x^{2}} x + 8 (- 14 x)$

Step 3.4.9

Multiply $- 14$ by $6$ .

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 140 x^{4} + \frac{- 84}{x^{2}} x + 8 (- 14 x)$

Step 3.4.10

Combine $\frac{- 84}{x^{2}}$ and $x$ .

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 140 x^{4} + \frac{- 84 x}{x^{2}} + 8 (- 14 x)$

Step 3.4.11

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

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Step 3.4.11.1

Factor $x$ out of $- 84 x$ .

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 140 x^{4} + \frac{x \cdot - 84}{x^{2}} + 8 (- 14 x)$

Step 3.4.11.2

Cancel the common factors.

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Step 3.4.11.2.1

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

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 140 x^{4} + \frac{x \cdot - 84}{x \cdot x} + 8 (- 14 x)$

Step 3.4.11.2.2

Cancel the common factor.

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 140 x^{4} + \frac{x \cdot - 84}{x \cdot x} + 8 (- 14 x)$

Step 3.4.11.2.3

Rewrite the expression.

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 140 x^{4} + \frac{- 84}{x} + 8 (- 14 x)$

Step 3.4.12

Move the negative in front of the fraction.

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 140 x^{4} - \frac{84}{x} + 8 (- 14 x)$

Step 3.4.13

Multiply $- 14$ by $8$ .

$(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 140 x^{4} - \frac{84}{x} - 112 x$

Step 3.4.14

To write $(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}})$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- 140 x^{4} + \frac{(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x}{x} - \frac{84}{x} - 112 x$

Step 3.4.15

Combine the numerators over the common denominator.

$- 140 x^{4} + \frac{(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x - 84}{x} - 112 x$

Step 3.4.16

To write $- 140 x^{4}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- 140 x^{4} \cdot \frac{x}{x} + \frac{(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x - 84}{x} - 112 x$

Step 3.4.17

Combine $- 140 x^{4}$ and $\frac{x}{x}$ .

$\frac{- 140 x^{4} x}{x} + \frac{(10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x - 84}{x} - 112 x$

Step 3.4.18

Combine the numerators over the common denominator.

$\frac{- 140 x^{4} x + (10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x - 84}{x} - 112 x$

Step 3.4.19

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

$\frac{- 140 (x^{1} x^{4}) + (10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x - 84}{x} - 112 x$

Step 3.4.20

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

$\frac{- 140 x^{1 + 4} + (10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x - 84}{x} - 112 x$

Step 3.4.21

Add $1$ and $4$ .

$\frac{- 140 x^{5} + (10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x - 84}{x} - 112 x$

Step 3.4.22

To write $- 112 x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{- 140 x^{5} + (10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x - 84}{x} - 112 x \cdot \frac{x}{x}$

Step 3.4.23

Combine $- 112 x$ and $\frac{x}{x}$ .

$\frac{- 140 x^{5} + (10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x - 84}{x} + \frac{- 112 x \cdot x}{x}$

Step 3.4.24

Combine the numerators over the common denominator.

$\frac{- 140 x^{5} + (10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x - 84 - 112 x \cdot x}{x}$

Step 3.4.25

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

$\frac{- 140 x^{5} + (10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x - 84 - 112 (x^{1} x)}{x}$

Step 3.4.26

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

$\frac{- 140 x^{5} + (10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x - 84 - 112 (x^{1} x^{1})}{x}$

Step 3.4.27

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

$\frac{- 140 x^{5} + (10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x - 84 - 112 x^{1 + 1}}{x}$

Step 3.4.28

Add $1$ and $1$ .

$\frac{- 140 x^{5} + (10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) x - 84 - 112 x^{2}}{x}$

Step 3.5

Reorder terms.

$\frac{- 140 x^{5} - 112 x^{2} + x (10 - 7 x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 84}{x}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

$\frac{- 140 x^{5} - 112 x^{2} + (x \cdot 10 + x (- 7 x^{2})) (30 x^{2} - \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.2

Move $10$ to the left of $x$ .

$\frac{- 140 x^{5} - 112 x^{2} + (10 \cdot x + x (- 7 x^{2})) (30 x^{2} - \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.3

Rewrite using the commutative property of multiplication.

$\frac{- 140 x^{5} - 112 x^{2} + (10 \cdot x - 7 x \cdot x^{2}) (30 x^{2} - \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.4

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

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Step 3.6.4.1

Move $x^{2}$ .

$\frac{- 140 x^{5} - 112 x^{2} + (10 x - 7 (x^{2} x)) (30 x^{2} - \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.4.2

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

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Step 3.6.4.2.1

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

$\frac{- 140 x^{5} - 112 x^{2} + (10 x - 7 (x^{2} x^{1})) (30 x^{2} - \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.4.2.2

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

$\frac{- 140 x^{5} - 112 x^{2} + (10 x - 7 x^{2 + 1}) (30 x^{2} - \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.4.3

Add $2$ and $1$ .

$\frac{- 140 x^{5} - 112 x^{2} + (10 x - 7 x^{3}) (30 x^{2} - \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.5

Expand $(10 x - 7 x^{3}) (30 x^{2} - \frac{12}{x^{3}})$ using the FOIL Method.

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Step 3.6.5.1

Apply the distributive property.

$\frac{- 140 x^{5} - 112 x^{2} + 10 x (30 x^{2} - \frac{12}{x^{3}}) - 7 x^{3} (30 x^{2} - \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.5.2

Apply the distributive property.

$\frac{- 140 x^{5} - 112 x^{2} + 10 x (30 x^{2}) + 10 x (- \frac{12}{x^{3}}) - 7 x^{3} (30 x^{2} - \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.5.3

Apply the distributive property.

$\frac{- 140 x^{5} - 112 x^{2} + 10 x (30 x^{2}) + 10 x (- \frac{12}{x^{3}}) - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6

Simplify each term.

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Step 3.6.6.1

Rewrite using the commutative property of multiplication.

$\frac{- 140 x^{5} - 112 x^{2} + 10 \cdot 30 x \cdot x^{2} + 10 x (- \frac{12}{x^{3}}) - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.2

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

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Step 3.6.6.2.1

Move $x^{2}$ .

$\frac{- 140 x^{5} - 112 x^{2} + 10 \cdot 30 (x^{2} x) + 10 x (- \frac{12}{x^{3}}) - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.2.2

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

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Step 3.6.6.2.2.1

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

$\frac{- 140 x^{5} - 112 x^{2} + 10 \cdot 30 (x^{2} x^{1}) + 10 x (- \frac{12}{x^{3}}) - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.2.2.2

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

$\frac{- 140 x^{5} - 112 x^{2} + 10 \cdot 30 x^{2 + 1} + 10 x (- \frac{12}{x^{3}}) - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.2.3

Add $2$ and $1$ .

$\frac{- 140 x^{5} - 112 x^{2} + 10 \cdot 30 x^{3} + 10 x (- \frac{12}{x^{3}}) - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.3

Multiply $10$ by $30$ .

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} + 10 x (- \frac{12}{x^{3}}) - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.4

Cancel the common factor of $x$ .

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Step 3.6.6.4.1

Move the leading negative in $- \frac{12}{x^{3}}$ into the numerator.

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} + 10 x \frac{- 12}{x^{3}} - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.4.2

Factor $x$ out of $10 x$ .

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} + x \cdot 10 \frac{- 12}{x^{3}} - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.4.3

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

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} + x \cdot 10 \frac{- 12}{x \cdot x^{2}} - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.4.4

Cancel the common factor.

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} + x \cdot 10 \frac{- 12}{x \cdot x^{2}} - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.4.5

Rewrite the expression.

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} + 10 \frac{- 12}{x^{2}} - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.5

Combine $10$ and $\frac{- 12}{x^{2}}$ .

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} + \frac{10 \cdot - 12}{x^{2}} - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.6

Multiply $10$ by $- 12$ .

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} + \frac{- 120}{x^{2}} - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.7

Move the negative in front of the fraction.

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}} - 7 x^{3} (30 x^{2}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.8

Rewrite using the commutative property of multiplication.

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}} - 7 \cdot 30 x^{3} x^{2} - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.9

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 3.6.6.9.1

Move $x^{2}$ .

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}} - 7 \cdot 30 (x^{2} x^{3}) - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.9.2

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

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}} - 7 \cdot 30 x^{2 + 3} - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.9.3

Add $2$ and $3$ .

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}} - 7 \cdot 30 x^{5} - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.10

Multiply $- 7$ by $30$ .

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}} - 210 x^{5} - 7 x^{3} (- \frac{12}{x^{3}}) - 84}{x}$

Step 3.6.6.11

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

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Step 3.6.6.11.1

Move the leading negative in $- \frac{12}{x^{3}}$ into the numerator.

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}} - 210 x^{5} - 7 x^{3} \frac{- 12}{x^{3}} - 84}{x}$

Step 3.6.6.11.2

Factor $x^{3}$ out of $- 7 x^{3}$ .

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}} - 210 x^{5} + x^{3} \cdot - 7 \frac{- 12}{x^{3}} - 84}{x}$

Step 3.6.6.11.3

Cancel the common factor.

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}} - 210 x^{5} + x^{3} \cdot - 7 \frac{- 12}{x^{3}} - 84}{x}$

Step 3.6.6.11.4

Rewrite the expression.

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}} - 210 x^{5} - 7 \cdot - 12 - 84}{x}$

Step 3.6.6.12

Multiply $- 7$ by $- 12$ .

$\frac{- 140 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}} - 210 x^{5} + 84 - 84}{x}$

Step 3.6.7

Subtract $210 x^{5}$ from $- 140 x^{5}$ .

$\frac{- 350 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}} + 84 - 84}{x}$

Step 3.6.8

Subtract $84$ from $84$ .

$\frac{- 350 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}} + 0}{x}$

Step 3.6.9

Add $- 350 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}}$ and $0$ .

$\frac{- 350 x^{5} - 112 x^{2} + 300 x^{3} - \frac{120}{x^{2}}}{x}$

Step 3.6.10

Reorder terms.

$\frac{- 350 x^{5} + 300 x^{3} - 112 x^{2} - \frac{120}{x^{2}}}{x}$

Step 3.6.11

Factor $2$ out of $- 350 x^{5} + 300 x^{3} - 112 x^{2} - \frac{120}{x^{2}}$ .

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Step 3.6.11.1

Factor $2$ out of $- 350 x^{5}$ .

$\frac{2 (- 175 x^{5}) + 300 x^{3} - 112 x^{2} - \frac{120}{x^{2}}}{x}$

Step 3.6.11.2

Factor $2$ out of $300 x^{3}$ .

$\frac{2 (- 175 x^{5}) + 2 (150 x^{3}) - 112 x^{2} - \frac{120}{x^{2}}}{x}$

Step 3.6.11.3

Factor $2$ out of $- 112 x^{2}$ .

$\frac{2 (- 175 x^{5}) + 2 (150 x^{3}) + 2 (- 56 x^{2}) - \frac{120}{x^{2}}}{x}$

Step 3.6.11.4

Factor $2$ out of $- \frac{120}{x^{2}}$ .

$\frac{2 (- 175 x^{5}) + 2 (150 x^{3}) + 2 (- 56 x^{2}) + 2 (- \frac{60}{x^{2}})}{x}$

Step 3.6.11.5

Factor $2$ out of $2 (- 175 x^{5}) + 2 (150 x^{3})$ .

$\frac{2 (- 175 x^{5} + 150 x^{3}) + 2 (- 56 x^{2}) + 2 (- \frac{60}{x^{2}})}{x}$

Step 3.6.11.6

Factor $2$ out of $2 (- 175 x^{5} + 150 x^{3}) + 2 (- 56 x^{2})$ .

$\frac{2 (- 175 x^{5} + 150 x^{3} - 56 x^{2}) + 2 (- \frac{60}{x^{2}})}{x}$

Step 3.6.11.7

Factor $2$ out of $2 (- 175 x^{5} + 150 x^{3} - 56 x^{2}) + 2 (- \frac{60}{x^{2}})$ .

$\frac{2 (- 175 x^{5} + 150 x^{3} - 56 x^{2} - \frac{60}{x^{2}})}{x}$

Step 3.6.12

To write $- 175 x^{5}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{2 (150 x^{3} - 56 x^{2} - 175 x^{5} \cdot \frac{x^{2}}{x^{2}} - \frac{60}{x^{2}})}{x}$

Step 3.6.13

Combine $- 175 x^{5}$ and $\frac{x^{2}}{x^{2}}$ .

$\frac{2 (150 x^{3} - 56 x^{2} + \frac{- 175 x^{5} x^{2}}{x^{2}} - \frac{60}{x^{2}})}{x}$

Step 3.6.14

Combine the numerators over the common denominator.

$\frac{2 (150 x^{3} - 56 x^{2} + \frac{- 175 x^{5} x^{2} - 60}{x^{2}})}{x}$

Step 3.6.15

Simplify the numerator.

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Step 3.6.15.1

Factor $5$ out of $- 175 x^{5} x^{2} - 60$ .

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Step 3.6.15.1.1

Factor $5$ out of $- 175 x^{5} x^{2}$ .

$\frac{2 (150 x^{3} - 56 x^{2} + \frac{5 (- 35 x^{5} x^{2}) - 60}{x^{2}})}{x}$

Step 3.6.15.1.2

Factor $5$ out of $- 60$ .

$\frac{2 (150 x^{3} - 56 x^{2} + \frac{5 (- 35 x^{5} x^{2}) + 5 (- 12)}{x^{2}})}{x}$

Step 3.6.15.1.3

Factor $5$ out of $5 (- 35 x^{5} x^{2}) + 5 (- 12)$ .

$\frac{2 (150 x^{3} - 56 x^{2} + \frac{5 (- 35 x^{5} x^{2} - 12)}{x^{2}})}{x}$

Step 3.6.15.2

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

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Step 3.6.15.2.1

Move $x^{2}$ .

$\frac{2 (150 x^{3} - 56 x^{2} + \frac{5 (- 35 (x^{2} x^{5}) - 12)}{x^{2}})}{x}$

Step 3.6.15.2.2

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

$\frac{2 (150 x^{3} - 56 x^{2} + \frac{5 (- 35 x^{2 + 5} - 12)}{x^{2}})}{x}$

Step 3.6.15.2.3

Add $2$ and $5$ .

$\frac{2 (150 x^{3} - 56 x^{2} + \frac{5 (- 35 x^{7} - 12)}{x^{2}})}{x}$

Step 3.6.16

To write $150 x^{3}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{2 (- 56 x^{2} + \frac{150 x^{3} x^{2}}{x^{2}} + \frac{5 (- 35 x^{7} - 12)}{x^{2}})}{x}$

Step 3.6.17

Combine the numerators over the common denominator.

$\frac{2 (- 56 x^{2} + \frac{150 x^{3} x^{2} + 5 (- 35 x^{7} - 12)}{x^{2}})}{x}$

Step 3.6.18

Simplify the numerator.

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Step 3.6.18.1

Factor $5$ out of $150 x^{3} x^{2} + 5 (- 35 x^{7} - 12)$ .

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Step 3.6.18.1.1

Factor $5$ out of $150 x^{3} x^{2}$ .

$\frac{2 (- 56 x^{2} + \frac{5 (30 x^{3} x^{2}) + 5 (- 35 x^{7} - 12)}{x^{2}})}{x}$

Step 3.6.18.1.2

Factor $5$ out of $5 (30 x^{3} x^{2}) + 5 (- 35 x^{7} - 12)$ .

$\frac{2 (- 56 x^{2} + \frac{5 (30 x^{3} x^{2} - 35 x^{7} - 12)}{x^{2}})}{x}$

Step 3.6.18.2

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 3.6.18.2.1

Move $x^{2}$ .

$\frac{2 (- 56 x^{2} + \frac{5 (30 (x^{2} x^{3}) - 35 x^{7} - 12)}{x^{2}})}{x}$

Step 3.6.18.2.2

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

$\frac{2 (- 56 x^{2} + \frac{5 (30 x^{2 + 3} - 35 x^{7} - 12)}{x^{2}})}{x}$

Step 3.6.18.2.3

Add $2$ and $3$ .

$\frac{2 (- 56 x^{2} + \frac{5 (30 x^{5} - 35 x^{7} - 12)}{x^{2}})}{x}$

Step 3.6.18.3

Reorder terms.

$\frac{2 (- 56 x^{2} + \frac{5 (- 35 x^{7} + 30 x^{5} - 12)}{x^{2}})}{x}$

Step 3.6.19

To write $- 56 x^{2}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{2 (- 56 x^{2} \cdot \frac{x^{2}}{x^{2}} + \frac{5 (- 35 x^{7} + 30 x^{5} - 12)}{x^{2}})}{x}$

Step 3.6.20

Combine $- 56 x^{2}$ and $\frac{x^{2}}{x^{2}}$ .

$\frac{2 (\frac{- 56 x^{2} x^{2}}{x^{2}} + \frac{5 (- 35 x^{7} + 30 x^{5} - 12)}{x^{2}})}{x}$

Step 3.6.21

Combine the numerators over the common denominator.

$\frac{2 \frac{- 56 x^{2} x^{2} + 5 (- 35 x^{7} + 30 x^{5} - 12)}{x^{2}}}{x}$

Step 3.6.22

Simplify the numerator.

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Step 3.6.22.1

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

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Step 3.6.22.1.1

Move $x^{2}$ .

$\frac{2 \frac{- 56 (x^{2} x^{2}) + 5 (- 35 x^{7} + 30 x^{5} - 12)}{x^{2}}}{x}$

Step 3.6.22.1.2

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

$\frac{2 \frac{- 56 x^{2 + 2} + 5 (- 35 x^{7} + 30 x^{5} - 12)}{x^{2}}}{x}$

Step 3.6.22.1.3

Add $2$ and $2$ .

$\frac{2 \frac{- 56 x^{4} + 5 (- 35 x^{7} + 30 x^{5} - 12)}{x^{2}}}{x}$

Step 3.6.22.2

Apply the distributive property.

$\frac{2 \frac{- 56 x^{4} + 5 (- 35 x^{7}) + 5 (30 x^{5}) + 5 \cdot - 12}{x^{2}}}{x}$

Step 3.6.22.3

Simplify.

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Step 3.6.22.3.1

Multiply $- 35$ by $5$ .

$\frac{2 \frac{- 56 x^{4} - 175 x^{7} + 5 (30 x^{5}) + 5 \cdot - 12}{x^{2}}}{x}$

Step 3.6.22.3.2

Multiply $30$ by $5$ .

$\frac{2 \frac{- 56 x^{4} - 175 x^{7} + 150 x^{5} + 5 \cdot - 12}{x^{2}}}{x}$

Step 3.6.22.3.3

Multiply $5$ by $- 12$ .

$\frac{2 \frac{- 56 x^{4} - 175 x^{7} + 150 x^{5} - 60}{x^{2}}}{x}$

Step 3.6.22.4

Reorder terms.

$\frac{2 \frac{- 175 x^{7} + 150 x^{5} - 56 x^{4} - 60}{x^{2}}}{x}$

Step 3.7

Combine $2$ and $\frac{- 175 x^{7} + 150 x^{5} - 56 x^{4} - 60}{x^{2}}$ .

$\frac{\frac{2 (- 175 x^{7} + 150 x^{5} - 56 x^{4} - 60)}{x^{2}}}{x}$

Step 3.8

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 (- 175 x^{7} + 150 x^{5} - 56 x^{4} - 60)}{x^{2}} \cdot \frac{1}{x}$

Step 3.9

Combine.

$\frac{2 (- 175 x^{7} + 150 x^{5} - 56 x^{4} - 60) \cdot 1}{x^{2} x}$

Step 3.10

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

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Step 3.10.1

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

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Step 3.10.1.1

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

$\frac{2 (- 175 x^{7} + 150 x^{5} - 56 x^{4} - 60) \cdot 1}{x^{2} x^{1}}$

Step 3.10.1.2

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

$\frac{2 (- 175 x^{7} + 150 x^{5} - 56 x^{4} - 60) \cdot 1}{x^{2 + 1}}$

Step 3.10.2

Add $2$ and $1$ .

$\frac{2 (- 175 x^{7} + 150 x^{5} - 56 x^{4} - 60) \cdot 1}{x^{3}}$

Step 3.11

Multiply $2 (- 175 x^{7} + 150 x^{5} - 56 x^{4} - 60)$ by $1$ .

$\frac{2 (- 175 x^{7} + 150 x^{5} - 56 x^{4} - 60)}{x^{3}}$

Step 3.12

Factor $- 1$ out of $- 175 x^{7}$ .

$\frac{2 (- (175 x^{7}) + 150 x^{5} - 56 x^{4} - 60)}{x^{3}}$

Step 3.13

Factor $- 1$ out of $150 x^{5}$ .

$\frac{2 (- (175 x^{7}) - (- 150 x^{5}) - 56 x^{4} - 60)}{x^{3}}$

Step 3.14

Factor $- 1$ out of $- (175 x^{7}) - (- 150 x^{5})$ .

$\frac{2 (- (175 x^{7} - 150 x^{5}) - 56 x^{4} - 60)}{x^{3}}$

Step 3.15

Factor $- 1$ out of $- 56 x^{4}$ .

$\frac{2 (- (175 x^{7} - 150 x^{5}) - (56 x^{4}) - 60)}{x^{3}}$

Step 3.16

Factor $- 1$ out of $- (175 x^{7} - 150 x^{5}) - (56 x^{4})$ .

$\frac{2 (- (175 x^{7} - 150 x^{5} + 56 x^{4}) - 60)}{x^{3}}$

Step 3.17

Rewrite $- 60$ as $- 1 (60)$ .

$\frac{2 (- (175 x^{7} - 150 x^{5} + 56 x^{4}) - 1 (60))}{x^{3}}$

Step 3.18

Factor $- 1$ out of $- (175 x^{7} - 150 x^{5} + 56 x^{4}) - 1 (60)$ .

$\frac{2 (- (175 x^{7} - 150 x^{5} + 56 x^{4} + 60))}{x^{3}}$

Step 3.19

Rewrite $- (175 x^{7} - 150 x^{5} + 56 x^{4} + 60)$ as $- 1 (175 x^{7} - 150 x^{5} + 56 x^{4} + 60)$ .

$\frac{2 (- 1 (175 x^{7} - 150 x^{5} + 56 x^{4} + 60))}{x^{3}}$

Step 3.20

Move the negative in front of the fraction.

$- \frac{2 (175 x^{7} - 150 x^{5} + 56 x^{4} + 60)}{x^{3}}$