Calculus Examples

Derivar $y = (\frac{5}{x^{4}} + \frac{3}{x^{3}}) (2 x^{5} - 4 x^{2})$

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

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $2 x^{5} - 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [2 x^{5}] + \frac{d}{d x} [- 4 x^{2}]$ .

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

Step 2.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{5}$ with respect to $x$ is $2 \frac{d}{d x} [x^{5}]$ .

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

Step 2.3

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

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

Step 2.4

Multiply $5$ by $2$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Multiply $2$ by $- 4$ .

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) \frac{d}{d x} [\frac{5}{x^{4}} + \frac{3}{x^{3}}]$

Step 2.8

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

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (\frac{d}{d x} [\frac{5}{x^{4}}] + \frac{d}{d x} [\frac{3}{x^{3}}])$

Step 2.9

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x^{4}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (5 \frac{d}{d x} [\frac{1}{x^{4}}] + \frac{d}{d x} [\frac{3}{x^{3}}])$

Step 2.10

Apply basic rules of exponents.

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Step 2.10.1

Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (5 \frac{d}{d x} [{(x^{4})}^{- 1}] + \frac{d}{d x} [\frac{3}{x^{3}}])$

Step 2.10.2

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 2.10.2.1

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

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (5 \frac{d}{d x} [x^{4 \cdot - 1}] + \frac{d}{d x} [\frac{3}{x^{3}}])$

Step 2.10.2.2

Multiply $4$ by $- 1$ .

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (5 \frac{d}{d x} [x^{- 4}] + \frac{d}{d x} [\frac{3}{x^{3}}])$

Step 2.11

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

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (5 (- 4 x^{- 5}) + \frac{d}{d x} [\frac{3}{x^{3}}])$

Step 2.12

Multiply $- 4$ by $5$ .

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (- 20 x^{- 5} + \frac{d}{d x} [\frac{3}{x^{3}}])$

Step 2.13

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x^{3}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (- 20 x^{- 5} + 3 \frac{d}{d x} [\frac{1}{x^{3}}])$

Step 2.14

Apply basic rules of exponents.

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Step 2.14.1

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

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (- 20 x^{- 5} + 3 \frac{d}{d x} [{(x^{3})}^{- 1}])$

Step 2.14.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 2.14.2.1

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

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (- 20 x^{- 5} + 3 \frac{d}{d x} [x^{3 \cdot - 1}])$

Step 2.14.2.2

Multiply $3$ by $- 1$ .

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (- 20 x^{- 5} + 3 \frac{d}{d x} [x^{- 3}])$

Step 2.15

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

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (- 20 x^{- 5} + 3 (- 3 x^{- 4}))$

Step 2.16

Multiply $- 3$ by $3$ .

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (- 20 x^{- 5} - 9 x^{- 4})$

Step 3

Simplify.

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Step 3.1

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

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (- 20 \frac{1}{x^{5}} - 9 x^{- 4})$

Step 3.2

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

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (- 20 \frac{1}{x^{5}} - 9 \frac{1}{x^{4}})$

Step 3.3

Combine terms.

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Step 3.3.1

Combine $- 20$ and $\frac{1}{x^{5}}$ .

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (\frac{- 20}{x^{5}} - 9 \frac{1}{x^{4}})$

Step 3.3.2

Move the negative in front of the fraction.

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (- \frac{20}{x^{5}} - 9 \frac{1}{x^{4}})$

Step 3.3.3

Combine $- 9$ and $\frac{1}{x^{4}}$ .

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (- \frac{20}{x^{5}} + \frac{- 9}{x^{4}})$

Step 3.3.4

Move the negative in front of the fraction.

$(\frac{5}{x^{4}} + \frac{3}{x^{3}}) (10 x^{4} - 8 x) + (2 x^{5} - 4 x^{2}) (- \frac{20}{x^{5}} - \frac{9}{x^{4}})$

Step 3.4

Reorder terms.

$(10 x^{4} - 8 x) (\frac{5}{x^{4}} + \frac{3}{x^{3}}) + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5

Simplify each term.

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Step 3.5.1

Expand $(10 x^{4} - 8 x) (\frac{5}{x^{4}} + \frac{3}{x^{3}})$ using the FOIL Method.

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Step 3.5.1.1

Apply the distributive property.

$10 x^{4} (\frac{5}{x^{4}} + \frac{3}{x^{3}}) - 8 x (\frac{5}{x^{4}} + \frac{3}{x^{3}}) + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.1.2

Apply the distributive property.

$10 x^{4} \frac{5}{x^{4}} + 10 x^{4} \frac{3}{x^{3}} - 8 x (\frac{5}{x^{4}} + \frac{3}{x^{3}}) + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.1.3

Apply the distributive property.

$10 x^{4} \frac{5}{x^{4}} + 10 x^{4} \frac{3}{x^{3}} - 8 x \frac{5}{x^{4}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

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

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Step 3.5.2.1.1

Factor $x^{4}$ out of $10 x^{4}$ .

$x^{4} \cdot 10 \frac{5}{x^{4}} + 10 x^{4} \frac{3}{x^{3}} - 8 x \frac{5}{x^{4}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.1.2

Cancel the common factor.

$x^{4} \cdot 10 \frac{5}{x^{4}} + 10 x^{4} \frac{3}{x^{3}} - 8 x \frac{5}{x^{4}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.1.3

Rewrite the expression.

$10 \cdot 5 + 10 x^{4} \frac{3}{x^{3}} - 8 x \frac{5}{x^{4}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.2

Multiply $10$ by $5$ .

$50 + 10 x^{4} \frac{3}{x^{3}} - 8 x \frac{5}{x^{4}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.3

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

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Step 3.5.2.3.1

Factor $x^{3}$ out of $10 x^{4}$ .

$50 + x^{3} (10 x) \frac{3}{x^{3}} - 8 x \frac{5}{x^{4}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.3.2

Cancel the common factor.

$50 + x^{3} (10 x) \frac{3}{x^{3}} - 8 x \frac{5}{x^{4}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.3.3

Rewrite the expression.

$50 + 10 x \cdot 3 - 8 x \frac{5}{x^{4}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.4

Multiply $3$ by $10$ .

$50 + 30 x - 8 x \frac{5}{x^{4}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.5

Cancel the common factor of $x$ .

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Step 3.5.2.5.1

Factor $x$ out of $- 8 x$ .

$50 + 30 x + x \cdot - 8 \frac{5}{x^{4}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.5.2

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

$50 + 30 x + x \cdot - 8 \frac{5}{x \cdot x^{3}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.5.3

Cancel the common factor.

$50 + 30 x + x \cdot - 8 \frac{5}{x \cdot x^{3}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.5.4

Rewrite the expression.

$50 + 30 x - 8 \frac{5}{x^{3}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.6

Combine $- 8$ and $\frac{5}{x^{3}}$ .

$50 + 30 x + \frac{- 8 \cdot 5}{x^{3}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.7

Multiply $- 8$ by $5$ .

$50 + 30 x + \frac{- 40}{x^{3}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.8

Move the negative in front of the fraction.

$50 + 30 x - \frac{40}{x^{3}} - 8 x \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.9

Cancel the common factor of $x$ .

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Step 3.5.2.9.1

Factor $x$ out of $- 8 x$ .

$50 + 30 x - \frac{40}{x^{3}} + x \cdot - 8 \frac{3}{x^{3}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.9.2

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

$50 + 30 x - \frac{40}{x^{3}} + x \cdot - 8 \frac{3}{x \cdot x^{2}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.9.3

Cancel the common factor.

$50 + 30 x - \frac{40}{x^{3}} + x \cdot - 8 \frac{3}{x \cdot x^{2}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.9.4

Rewrite the expression.

$50 + 30 x - \frac{40}{x^{3}} - 8 \frac{3}{x^{2}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.10

Combine $- 8$ and $\frac{3}{x^{2}}$ .

$50 + 30 x - \frac{40}{x^{3}} + \frac{- 8 \cdot 3}{x^{2}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.11

Multiply $- 8$ by $3$ .

$50 + 30 x - \frac{40}{x^{3}} + \frac{- 24}{x^{2}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.2.12

Move the negative in front of the fraction.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} + (- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$

Step 3.5.3

Expand $(- \frac{20}{x^{5}} - \frac{9}{x^{4}}) (2 x^{5} - 4 x^{2})$ using the FOIL Method.

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Step 3.5.3.1

Apply the distributive property.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - \frac{20}{x^{5}} (2 x^{5} - 4 x^{2}) - \frac{9}{x^{4}} (2 x^{5} - 4 x^{2})$

Step 3.5.3.2

Apply the distributive property.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - \frac{20}{x^{5}} (2 x^{5}) - \frac{20}{x^{5}} (- 4 x^{2}) - \frac{9}{x^{4}} (2 x^{5} - 4 x^{2})$

Step 3.5.3.3

Apply the distributive property.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - \frac{20}{x^{5}} (2 x^{5}) - \frac{20}{x^{5}} (- 4 x^{2}) - \frac{9}{x^{4}} (2 x^{5}) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4

Simplify each term.

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Step 3.5.4.1

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

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Step 3.5.4.1.1

Move the leading negative in $- \frac{20}{x^{5}}$ into the numerator.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} + \frac{- 20}{x^{5}} (2 x^{5}) - \frac{20}{x^{5}} (- 4 x^{2}) - \frac{9}{x^{4}} (2 x^{5}) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.1.2

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

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} + \frac{- 20}{x^{5}} (x^{5} \cdot 2) - \frac{20}{x^{5}} (- 4 x^{2}) - \frac{9}{x^{4}} (2 x^{5}) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.1.3

Cancel the common factor.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} + \frac{- 20}{x^{5}} (x^{5} \cdot 2) - \frac{20}{x^{5}} (- 4 x^{2}) - \frac{9}{x^{4}} (2 x^{5}) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.1.4

Rewrite the expression.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 20 \cdot 2 - \frac{20}{x^{5}} (- 4 x^{2}) - \frac{9}{x^{4}} (2 x^{5}) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.2

Multiply $- 20$ by $2$ .

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 - \frac{20}{x^{5}} (- 4 x^{2}) - \frac{9}{x^{4}} (2 x^{5}) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.3

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

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Step 3.5.4.3.1

Move the leading negative in $- \frac{20}{x^{5}}$ into the numerator.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{- 20}{x^{5}} (- 4 x^{2}) - \frac{9}{x^{4}} (2 x^{5}) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.3.2

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

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{- 20}{x^{2} x^{3}} (- 4 x^{2}) - \frac{9}{x^{4}} (2 x^{5}) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.3.3

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

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{- 20}{x^{2} x^{3}} (x^{2} \cdot - 4) - \frac{9}{x^{4}} (2 x^{5}) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.3.4

Cancel the common factor.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{- 20}{x^{2} x^{3}} (x^{2} \cdot - 4) - \frac{9}{x^{4}} (2 x^{5}) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.3.5

Rewrite the expression.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{- 20}{x^{3}} \cdot - 4 - \frac{9}{x^{4}} (2 x^{5}) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.4

Combine $\frac{- 20}{x^{3}}$ and $- 4$ .

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{- 20 \cdot - 4}{x^{3}} - \frac{9}{x^{4}} (2 x^{5}) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.5

Multiply $- 20$ by $- 4$ .

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{80}{x^{3}} - \frac{9}{x^{4}} (2 x^{5}) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.6

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

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Step 3.5.4.6.1

Move the leading negative in $- \frac{9}{x^{4}}$ into the numerator.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{80}{x^{3}} + \frac{- 9}{x^{4}} (2 x^{5}) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.6.2

Factor $x^{4}$ out of $2 x^{5}$ .

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{80}{x^{3}} + \frac{- 9}{x^{4}} (x^{4} (2 x)) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.6.3

Cancel the common factor.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{80}{x^{3}} + \frac{- 9}{x^{4}} (x^{4} (2 x)) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.6.4

Rewrite the expression.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{80}{x^{3}} - 9 (2 x) - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.7

Multiply $2$ by $- 9$ .

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{80}{x^{3}} - 18 x - \frac{9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.8

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

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Step 3.5.4.8.1

Move the leading negative in $- \frac{9}{x^{4}}$ into the numerator.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{80}{x^{3}} - 18 x + \frac{- 9}{x^{4}} (- 4 x^{2})$

Step 3.5.4.8.2

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

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{80}{x^{3}} - 18 x + \frac{- 9}{x^{2} x^{2}} (- 4 x^{2})$

Step 3.5.4.8.3

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

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{80}{x^{3}} - 18 x + \frac{- 9}{x^{2} x^{2}} (x^{2} \cdot - 4)$

Step 3.5.4.8.4

Cancel the common factor.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{80}{x^{3}} - 18 x + \frac{- 9}{x^{2} x^{2}} (x^{2} \cdot - 4)$

Step 3.5.4.8.5

Rewrite the expression.

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{80}{x^{3}} - 18 x + \frac{- 9}{x^{2}} \cdot - 4$

Step 3.5.4.9

Combine $\frac{- 9}{x^{2}}$ and $- 4$ .

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{80}{x^{3}} - 18 x + \frac{- 9 \cdot - 4}{x^{2}}$

Step 3.5.4.10

Multiply $- 9$ by $- 4$ .

$50 + 30 x - \frac{40}{x^{3}} - \frac{24}{x^{2}} - 40 + \frac{80}{x^{3}} - 18 x + \frac{36}{x^{2}}$

Step 3.6

Combine the numerators over the common denominator.

$50 + 30 x - 40 - 18 x + \frac{- 40 + 80}{x^{3}} + \frac{- 24 + 36}{x^{2}}$

Step 3.7

Add $- 40$ and $80$ .

$50 + 30 x - 40 - 18 x + \frac{40}{x^{3}} + \frac{- 24 + 36}{x^{2}}$

Step 3.8

Add $- 24$ and $36$ .

$50 + 30 x - 40 - 18 x + \frac{40}{x^{3}} + \frac{12}{x^{2}}$

Step 3.9

Subtract $40$ from $50$ .

$30 x + 10 - 18 x + \frac{40}{x^{3}} + \frac{12}{x^{2}}$

Step 3.10

Subtract $18 x$ from $30 x$ .

$12 x + 10 + \frac{40}{x^{3}} + \frac{12}{x^{2}}$