Calculus Examples

$y = (5 + 3 x^{- 2}) (4 x^{5} + 6 x^{3} + 10)$

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 $f (x) = 5 + 3 x^{- 2}$ and $g (x) = 4 x^{5} + 6 x^{3} + 10$ .

$(5 + 3 x^{- 2}) \frac{d}{d x} [4 x^{5} + 6 x^{3} + 10] + (4 x^{5} + 6 x^{3} + 10) \frac{d}{d x} [5 + 3 x^{- 2}]$

Step 2

By the Sum Rule, the derivative of $4 x^{5} + 6 x^{3} + 10$ with respect to $x$ is $\frac{d}{d x} [4 x^{5}] + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [10]$ .

$(5 + 3 x^{- 2}) (\frac{d}{d x} [4 x^{5}] + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [10]) + (4 x^{5} + 6 x^{3} + 10) \frac{d}{d x} [5 + 3 x^{- 2}]$

Step 3

Evaluate $\frac{d}{d x} [4 x^{5}]$ .

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{5}$ with respect to $x$ is $4 \frac{d}{d x} [x^{5}]$ .

$(5 + 3 x^{- 2}) (4 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [10]) + (4 x^{5} + 6 x^{3} + 10) \frac{d}{d x} [5 + 3 x^{- 2}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$(5 + 3 x^{- 2}) (4 (5 x^{4}) + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [10]) + (4 x^{5} + 6 x^{3} + 10) \frac{d}{d x} [5 + 3 x^{- 2}]$

Step 3.3

Multiply $5$ by $4$ .

$(5 + 3 x^{- 2}) (20 x^{4} + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [10]) + (4 x^{5} + 6 x^{3} + 10) \frac{d}{d x} [5 + 3 x^{- 2}]$

Step 4

Evaluate $\frac{d}{d x} [6 x^{3}]$ .

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Step 4.1

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

$(5 + 3 x^{- 2}) (20 x^{4} + 6 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [10]) + (4 x^{5} + 6 x^{3} + 10) \frac{d}{d x} [5 + 3 x^{- 2}]$

Step 4.2

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

$(5 + 3 x^{- 2}) (20 x^{4} + 6 (3 x^{2}) + \frac{d}{d x} [10]) + (4 x^{5} + 6 x^{3} + 10) \frac{d}{d x} [5 + 3 x^{- 2}]$

Step 4.3

Multiply $3$ by $6$ .

$(5 + 3 x^{- 2}) (20 x^{4} + 18 x^{2} + \frac{d}{d x} [10]) + (4 x^{5} + 6 x^{3} + 10) \frac{d}{d x} [5 + 3 x^{- 2}]$

Step 5

Differentiate.

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Step 5.1

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

$(5 + 3 x^{- 2}) (20 x^{4} + 18 x^{2} + 0) + (4 x^{5} + 6 x^{3} + 10) \frac{d}{d x} [5 + 3 x^{- 2}]$

Step 5.2

Add $20 x^{4} + 18 x^{2}$ and $0$ .

$(5 + 3 x^{- 2}) (20 x^{4} + 18 x^{2}) + (4 x^{5} + 6 x^{3} + 10) \frac{d}{d x} [5 + 3 x^{- 2}]$

Step 5.3

By the Sum Rule, the derivative of $5 + 3 x^{- 2}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [3 x^{- 2}]$ .

$(5 + 3 x^{- 2}) (20 x^{4} + 18 x^{2}) + (4 x^{5} + 6 x^{3} + 10) (\frac{d}{d x} [5] + \frac{d}{d x} [3 x^{- 2}])$

Step 5.4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$(5 + 3 x^{- 2}) (20 x^{4} + 18 x^{2}) + (4 x^{5} + 6 x^{3} + 10) (0 + \frac{d}{d x} [3 x^{- 2}])$

Step 6

Evaluate $\frac{d}{d x} [3 x^{- 2}]$ .

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Step 6.1

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

$(5 + 3 x^{- 2}) (20 x^{4} + 18 x^{2}) + (4 x^{5} + 6 x^{3} + 10) (0 + 3 \frac{d}{d x} [x^{- 2}])$

Step 6.2

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

$(5 + 3 x^{- 2}) (20 x^{4} + 18 x^{2}) + (4 x^{5} + 6 x^{3} + 10) (0 + 3 (- 2 x^{- 3}))$

Step 6.3

Multiply $- 2$ by $3$ .

$(5 + 3 x^{- 2}) (20 x^{4} + 18 x^{2}) + (4 x^{5} + 6 x^{3} + 10) (0 - 6 x^{- 3})$

Step 7

Simplify.

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Step 7.1

Subtract $6 x^{- 3}$ from $0$ .

$(5 + 3 x^{- 2}) (20 x^{4} + 18 x^{2}) + (4 x^{5} + 6 x^{3} + 10) (- 6 x^{- 3})$

Step 7.2

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

$(5 + 3 x^{- 2}) (20 x^{4} + 18 x^{2}) + (4 x^{5} + 6 x^{3} + 10) (- 6 \frac{1}{x^{3}})$

Step 7.3

Combine $- 6$ and $\frac{1}{x^{3}}$ .

$(5 + 3 x^{- 2}) (20 x^{4} + 18 x^{2}) + (4 x^{5} + 6 x^{3} + 10) \frac{- 6}{x^{3}}$

Step 7.4

Move the negative in front of the fraction.

$(5 + 3 x^{- 2}) (20 x^{4} + 18 x^{2}) + (4 x^{5} + 6 x^{3} + 10) (- \frac{6}{x^{3}})$

Step 8

Simplify.

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Step 8.1

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

$(5 + 3 \frac{1}{x^{2}}) (20 x^{4} + 18 x^{2}) + (4 x^{5} + 6 x^{3} + 10) (- \frac{6}{x^{3}})$

Step 8.2

Apply the distributive property.

$(5 + 3 \frac{1}{x^{2}}) (20 x^{4} + 18 x^{2}) + 4 x^{5} (- \frac{6}{x^{3}}) + 6 x^{3} (- \frac{6}{x^{3}}) + 10 (- \frac{6}{x^{3}})$

Step 8.3

Combine terms.

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Step 8.3.1

Combine $3$ and $\frac{1}{x^{2}}$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) + 4 x^{5} (- \frac{6}{x^{3}}) + 6 x^{3} (- \frac{6}{x^{3}}) + 10 (- \frac{6}{x^{3}})$

Step 8.3.2

Multiply $- 1$ by $4$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) - 4 x^{5} \frac{6}{x^{3}} + 6 x^{3} (- \frac{6}{x^{3}}) + 10 (- \frac{6}{x^{3}})$

Step 8.3.3

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

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) + x^{5} \frac{- 4 \cdot 6}{x^{3}} + 6 x^{3} (- \frac{6}{x^{3}}) + 10 (- \frac{6}{x^{3}})$

Step 8.3.4

Multiply $- 4$ by $6$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) + x^{5} \frac{- 24}{x^{3}} + 6 x^{3} (- \frac{6}{x^{3}}) + 10 (- \frac{6}{x^{3}})$

Step 8.3.5

Combine $x^{5}$ and $\frac{- 24}{x^{3}}$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) + \frac{x^{5} \cdot - 24}{x^{3}} + 6 x^{3} (- \frac{6}{x^{3}}) + 10 (- \frac{6}{x^{3}})$

Step 8.3.6

Move $- 24$ to the left of $x^{5}$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) + \frac{- 24 \cdot x^{5}}{x^{3}} + 6 x^{3} (- \frac{6}{x^{3}}) + 10 (- \frac{6}{x^{3}})$

Step 8.3.7

Cancel the common factor of $x^{5}$ and $x^{3}$ .

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Step 8.3.7.1

Factor $x^{3}$ out of $- 24 x^{5}$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) + \frac{x^{3} (- 24 x^{2})}{x^{3}} + 6 x^{3} (- \frac{6}{x^{3}}) + 10 (- \frac{6}{x^{3}})$

Step 8.3.7.2

Cancel the common factors.

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Step 8.3.7.2.1

Multiply by $1$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) + \frac{x^{3} (- 24 x^{2})}{x^{3} \cdot 1} + 6 x^{3} (- \frac{6}{x^{3}}) + 10 (- \frac{6}{x^{3}})$

Step 8.3.7.2.2

Cancel the common factor.

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) + \frac{x^{3} (- 24 x^{2})}{x^{3} \cdot 1} + 6 x^{3} (- \frac{6}{x^{3}}) + 10 (- \frac{6}{x^{3}})$

Step 8.3.7.2.3

Rewrite the expression.

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) + \frac{- 24 x^{2}}{1} + 6 x^{3} (- \frac{6}{x^{3}}) + 10 (- \frac{6}{x^{3}})$

Step 8.3.7.2.4

Divide $- 24 x^{2}$ by $1$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) - 24 x^{2} + 6 x^{3} (- \frac{6}{x^{3}}) + 10 (- \frac{6}{x^{3}})$

Step 8.3.8

Multiply $- 1$ by $6$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) - 24 x^{2} - 6 x^{3} \frac{6}{x^{3}} + 10 (- \frac{6}{x^{3}})$

Step 8.3.9

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

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) - 24 x^{2} + x^{3} \frac{- 6 \cdot 6}{x^{3}} + 10 (- \frac{6}{x^{3}})$

Step 8.3.10

Multiply $- 6$ by $6$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) - 24 x^{2} + x^{3} \frac{- 36}{x^{3}} + 10 (- \frac{6}{x^{3}})$

Step 8.3.11

Combine $x^{3}$ and $\frac{- 36}{x^{3}}$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) - 24 x^{2} + \frac{x^{3} \cdot - 36}{x^{3}} + 10 (- \frac{6}{x^{3}})$

Step 8.3.12

Move $- 36$ to the left of $x^{3}$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) - 24 x^{2} + \frac{- 36 \cdot x^{3}}{x^{3}} + 10 (- \frac{6}{x^{3}})$

Step 8.3.13

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

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Step 8.3.13.1

Cancel the common factor.

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) - 24 x^{2} + \frac{- 36 x^{3}}{x^{3}} + 10 (- \frac{6}{x^{3}})$

Step 8.3.13.2

Divide $- 36$ by $1$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) - 24 x^{2} - 36 + 10 (- \frac{6}{x^{3}})$

Step 8.3.14

Multiply $- 1$ by $10$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) - 24 x^{2} - 36 - 10 \frac{6}{x^{3}}$

Step 8.3.15

Combine $- 10$ and $\frac{6}{x^{3}}$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) - 24 x^{2} - 36 + \frac{- 10 \cdot 6}{x^{3}}$

Step 8.3.16

Multiply $- 10$ by $6$ .

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) - 24 x^{2} - 36 + \frac{- 60}{x^{3}}$

Step 8.3.17

Move the negative in front of the fraction.

$(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) - 24 x^{2} - 36 - \frac{60}{x^{3}}$

Step 8.3.18

To write $(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2})$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$\frac{(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) x^{3}}{x^{3}} - \frac{60}{x^{3}} - 24 x^{2} - 36$

Step 8.3.19

Combine the numerators over the common denominator.

$\frac{(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) x^{3} - 60}{x^{3}} - 24 x^{2} - 36$

Step 8.3.20

To write $- 24 x^{2}$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$\frac{(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) x^{3} - 60}{x^{3}} - 24 x^{2} \cdot \frac{x^{3}}{x^{3}} - 36$

Step 8.3.21

Combine $- 24 x^{2}$ and $\frac{x^{3}}{x^{3}}$ .

$\frac{(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) x^{3} - 60}{x^{3}} + \frac{- 24 x^{2} x^{3}}{x^{3}} - 36$

Step 8.3.22

Combine the numerators over the common denominator.

$\frac{(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) x^{3} - 60 - 24 x^{2} x^{3}}{x^{3}} - 36$

Step 8.3.23

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

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Step 8.3.23.1

Move $x^{3}$ .

$\frac{(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) x^{3} - 60 - 24 (x^{3} x^{2})}{x^{3}} - 36$

Step 8.3.23.2

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

$\frac{(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) x^{3} - 60 - 24 x^{3 + 2}}{x^{3}} - 36$

Step 8.3.23.3

Add $3$ and $2$ .

$\frac{(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) x^{3} - 60 - 24 x^{5}}{x^{3}} - 36$

Step 8.3.24

To write $- 36$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$\frac{(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) x^{3} - 60 - 24 x^{5}}{x^{3}} - 36 \cdot \frac{x^{3}}{x^{3}}$

Step 8.3.25

Combine $- 36$ and $\frac{x^{3}}{x^{3}}$ .

$\frac{(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) x^{3} - 60 - 24 x^{5}}{x^{3}} + \frac{- 36 x^{3}}{x^{3}}$

Step 8.3.26

Combine the numerators over the common denominator.

$\frac{(5 + \frac{3}{x^{2}}) (20 x^{4} + 18 x^{2}) x^{3} - 60 - 24 x^{5} - 36 x^{3}}{x^{3}}$

Step 8.4

Reorder terms.

$\frac{- 24 x^{5} - 36 x^{3} + x^{3} (20 x^{4} + 18 x^{2}) (5 + \frac{3}{x^{2}}) - 60}{x^{3}}$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Apply the distributive property.

$\frac{- 24 x^{5} - 36 x^{3} + (x^{3} (20 x^{4}) + x^{3} (18 x^{2})) (5 + \frac{3}{x^{2}}) - 60}{x^{3}}$

Step 8.5.2

Rewrite using the commutative property of multiplication.

$\frac{- 24 x^{5} - 36 x^{3} + (20 x^{3} x^{4} + x^{3} (18 x^{2})) (5 + \frac{3}{x^{2}}) - 60}{x^{3}}$

Step 8.5.3

Rewrite using the commutative property of multiplication.

$\frac{- 24 x^{5} - 36 x^{3} + (20 x^{3} x^{4} + 18 x^{3} x^{2}) (5 + \frac{3}{x^{2}}) - 60}{x^{3}}$

Step 8.5.4

Simplify each term.

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Step 8.5.4.1

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

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Step 8.5.4.1.1

Move $x^{4}$ .

$\frac{- 24 x^{5} - 36 x^{3} + (20 (x^{4} x^{3}) + 18 x^{3} x^{2}) (5 + \frac{3}{x^{2}}) - 60}{x^{3}}$

Step 8.5.4.1.2

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

$\frac{- 24 x^{5} - 36 x^{3} + (20 x^{4 + 3} + 18 x^{3} x^{2}) (5 + \frac{3}{x^{2}}) - 60}{x^{3}}$

Step 8.5.4.1.3

Add $4$ and $3$ .

$\frac{- 24 x^{5} - 36 x^{3} + (20 x^{7} + 18 x^{3} x^{2}) (5 + \frac{3}{x^{2}}) - 60}{x^{3}}$

Step 8.5.4.2

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

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Step 8.5.4.2.1

Move $x^{2}$ .

$\frac{- 24 x^{5} - 36 x^{3} + (20 x^{7} + 18 (x^{2} x^{3})) (5 + \frac{3}{x^{2}}) - 60}{x^{3}}$

Step 8.5.4.2.2

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

$\frac{- 24 x^{5} - 36 x^{3} + (20 x^{7} + 18 x^{2 + 3}) (5 + \frac{3}{x^{2}}) - 60}{x^{3}}$

Step 8.5.4.2.3

Add $2$ and $3$ .

$\frac{- 24 x^{5} - 36 x^{3} + (20 x^{7} + 18 x^{5}) (5 + \frac{3}{x^{2}}) - 60}{x^{3}}$

Step 8.5.5

Expand $(20 x^{7} + 18 x^{5}) (5 + \frac{3}{x^{2}})$ using the FOIL Method.

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Step 8.5.5.1

Apply the distributive property.

$\frac{- 24 x^{5} - 36 x^{3} + 20 x^{7} (5 + \frac{3}{x^{2}}) + 18 x^{5} (5 + \frac{3}{x^{2}}) - 60}{x^{3}}$

Step 8.5.5.2

Apply the distributive property.

$\frac{- 24 x^{5} - 36 x^{3} + 20 x^{7} \cdot 5 + 20 x^{7} \frac{3}{x^{2}} + 18 x^{5} (5 + \frac{3}{x^{2}}) - 60}{x^{3}}$

Step 8.5.5.3

Apply the distributive property.

$\frac{- 24 x^{5} - 36 x^{3} + 20 x^{7} \cdot 5 + 20 x^{7} \frac{3}{x^{2}} + 18 x^{5} \cdot 5 + 18 x^{5} \frac{3}{x^{2}} - 60}{x^{3}}$

Step 8.5.6

Simplify and combine like terms.

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Step 8.5.6.1

Simplify each term.

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Step 8.5.6.1.1

Multiply $5$ by $20$ .

$\frac{- 24 x^{5} - 36 x^{3} + 100 x^{7} + 20 x^{7} \frac{3}{x^{2}} + 18 x^{5} \cdot 5 + 18 x^{5} \frac{3}{x^{2}} - 60}{x^{3}}$

Step 8.5.6.1.2

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

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Step 8.5.6.1.2.1

Factor $x^{2}$ out of $20 x^{7}$ .

$\frac{- 24 x^{5} - 36 x^{3} + 100 x^{7} + x^{2} (20 x^{5}) \frac{3}{x^{2}} + 18 x^{5} \cdot 5 + 18 x^{5} \frac{3}{x^{2}} - 60}{x^{3}}$

Step 8.5.6.1.2.2

Cancel the common factor.

$\frac{- 24 x^{5} - 36 x^{3} + 100 x^{7} + x^{2} (20 x^{5}) \frac{3}{x^{2}} + 18 x^{5} \cdot 5 + 18 x^{5} \frac{3}{x^{2}} - 60}{x^{3}}$

Step 8.5.6.1.2.3

Rewrite the expression.

$\frac{- 24 x^{5} - 36 x^{3} + 100 x^{7} + 20 x^{5} \cdot 3 + 18 x^{5} \cdot 5 + 18 x^{5} \frac{3}{x^{2}} - 60}{x^{3}}$

Step 8.5.6.1.3

Multiply $3$ by $20$ .

$\frac{- 24 x^{5} - 36 x^{3} + 100 x^{7} + 60 x^{5} + 18 x^{5} \cdot 5 + 18 x^{5} \frac{3}{x^{2}} - 60}{x^{3}}$

Step 8.5.6.1.4

Multiply $5$ by $18$ .

$\frac{- 24 x^{5} - 36 x^{3} + 100 x^{7} + 60 x^{5} + 90 x^{5} + 18 x^{5} \frac{3}{x^{2}} - 60}{x^{3}}$

Step 8.5.6.1.5

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

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Step 8.5.6.1.5.1

Factor $x^{2}$ out of $18 x^{5}$ .

$\frac{- 24 x^{5} - 36 x^{3} + 100 x^{7} + 60 x^{5} + 90 x^{5} + x^{2} (18 x^{3}) \frac{3}{x^{2}} - 60}{x^{3}}$

Step 8.5.6.1.5.2

Cancel the common factor.

$\frac{- 24 x^{5} - 36 x^{3} + 100 x^{7} + 60 x^{5} + 90 x^{5} + x^{2} (18 x^{3}) \frac{3}{x^{2}} - 60}{x^{3}}$

Step 8.5.6.1.5.3

Rewrite the expression.

$\frac{- 24 x^{5} - 36 x^{3} + 100 x^{7} + 60 x^{5} + 90 x^{5} + 18 x^{3} \cdot 3 - 60}{x^{3}}$

Step 8.5.6.1.6

Multiply $3$ by $18$ .

$\frac{- 24 x^{5} - 36 x^{3} + 100 x^{7} + 60 x^{5} + 90 x^{5} + 54 x^{3} - 60}{x^{3}}$

Step 8.5.6.2

Add $60 x^{5}$ and $90 x^{5}$ .

$\frac{- 24 x^{5} - 36 x^{3} + 100 x^{7} + 150 x^{5} + 54 x^{3} - 60}{x^{3}}$

Step 8.5.7

Add $- 24 x^{5}$ and $150 x^{5}$ .

$\frac{126 x^{5} - 36 x^{3} + 100 x^{7} + 54 x^{3} - 60}{x^{3}}$

Step 8.5.8

Add $- 36 x^{3}$ and $54 x^{3}$ .

$\frac{126 x^{5} + 100 x^{7} + 18 x^{3} - 60}{x^{3}}$

Step 8.5.9

Reorder terms.

$\frac{100 x^{7} + 126 x^{5} + 18 x^{3} - 60}{x^{3}}$

Step 8.5.10

Factor $2$ out of $100 x^{7} + 126 x^{5} + 18 x^{3} - 60$ .

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Step 8.5.10.1

Factor $2$ out of $100 x^{7}$ .

$\frac{2 (50 x^{7}) + 126 x^{5} + 18 x^{3} - 60}{x^{3}}$

Step 8.5.10.2

Factor $2$ out of $126 x^{5}$ .

$\frac{2 (50 x^{7}) + 2 (63 x^{5}) + 18 x^{3} - 60}{x^{3}}$

Step 8.5.10.3

Factor $2$ out of $18 x^{3}$ .

$\frac{2 (50 x^{7}) + 2 (63 x^{5}) + 2 (9 x^{3}) - 60}{x^{3}}$

Step 8.5.10.4

Factor $2$ out of $- 60$ .

$\frac{2 (50 x^{7}) + 2 (63 x^{5}) + 2 (9 x^{3}) + 2 (- 30)}{x^{3}}$

Step 8.5.10.5

Factor $2$ out of $2 (50 x^{7}) + 2 (63 x^{5})$ .

$\frac{2 (50 x^{7} + 63 x^{5}) + 2 (9 x^{3}) + 2 (- 30)}{x^{3}}$

Step 8.5.10.6

Factor $2$ out of $2 (50 x^{7} + 63 x^{5}) + 2 (9 x^{3})$ .

$\frac{2 (50 x^{7} + 63 x^{5} + 9 x^{3}) + 2 (- 30)}{x^{3}}$

Step 8.5.10.7

Factor $2$ out of $2 (50 x^{7} + 63 x^{5} + 9 x^{3}) + 2 (- 30)$ .

$\frac{2 (50 x^{7} + 63 x^{5} + 9 x^{3} - 30)}{x^{3}}$