Calculus Examples

$p (y) = (y^{- 1} + y^{- 2}) (7 y^{- 3} - 9 y^{- 4})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = y^{- 1} + y^{- 2}$ and $g (y) = 7 y^{- 3} - 9 y^{- 4}$ .

$(y^{- 1} + y^{- 2}) \frac{d}{d y} [7 y^{- 3} - 9 y^{- 4}] + (7 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $7 y^{- 3} - 9 y^{- 4}$ with respect to $y$ is $\frac{d}{d y} [7 y^{- 3}] + \frac{d}{d y} [- 9 y^{- 4}]$ .

$(y^{- 1} + y^{- 2}) (\frac{d}{d y} [7 y^{- 3}] + \frac{d}{d y} [- 9 y^{- 4}]) + (7 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Step 2.2

Since $7$ is constant with respect to $y$ , the derivative of $7 y^{- 3}$ with respect to $y$ is $7 \frac{d}{d y} [y^{- 3}]$ .

$(y^{- 1} + y^{- 2}) (7 \frac{d}{d y} [y^{- 3}] + \frac{d}{d y} [- 9 y^{- 4}]) + (7 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Step 2.3

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

$(y^{- 1} + y^{- 2}) (7 (- 3 y^{- 4}) + \frac{d}{d y} [- 9 y^{- 4}]) + (7 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Step 2.4

Multiply $- 3$ by $7$ .

$(y^{- 1} + y^{- 2}) (- 21 y^{- 4} + \frac{d}{d y} [- 9 y^{- 4}]) + (7 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Step 2.5

Since $- 9$ is constant with respect to $y$ , the derivative of $- 9 y^{- 4}$ with respect to $y$ is $- 9 \frac{d}{d y} [y^{- 4}]$ .

$(y^{- 1} + y^{- 2}) (- 21 y^{- 4} - 9 \frac{d}{d y} [y^{- 4}]) + (7 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Step 2.6

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

$(y^{- 1} + y^{- 2}) (- 21 y^{- 4} - 9 (- 4 y^{- 5})) + (7 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Step 2.7

Multiply $- 4$ by $- 9$ .

$(y^{- 1} + y^{- 2}) (- 21 y^{- 4} + 36 y^{- 5}) + (7 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Step 2.8

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

$(y^{- 1} + y^{- 2}) (- 21 y^{- 4} + 36 y^{- 5}) + (7 y^{- 3} - 9 y^{- 4}) (\frac{d}{d y} [y^{- 1}] + \frac{d}{d y} [y^{- 2}])$

Step 2.9

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

$(y^{- 1} + y^{- 2}) (- 21 y^{- 4} + 36 y^{- 5}) + (7 y^{- 3} - 9 y^{- 4}) (- y^{- 2} + \frac{d}{d y} [y^{- 2}])$

Step 2.10

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

$(y^{- 1} + y^{- 2}) (- 21 y^{- 4} + 36 y^{- 5}) + (7 y^{- 3} - 9 y^{- 4}) (- y^{- 2} - 2 y^{- 3})$

Step 3

Simplify.

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Step 3.1

Reorder terms.

$(- 21 y^{- 4} + 36 y^{- 5}) (y^{- 1} + y^{- 2}) + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2

Simplify each term.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

$(- 21 \frac{1}{y^{4}} + 36 y^{- 5}) (y^{- 1} + y^{- 2}) + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.1.2

Combine $- 21$ and $\frac{1}{y^{4}}$ .

$(\frac{- 21}{y^{4}} + 36 y^{- 5}) (y^{- 1} + y^{- 2}) + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.1.3

Move the negative in front of the fraction.

$(- \frac{21}{y^{4}} + 36 y^{- 5}) (y^{- 1} + y^{- 2}) + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.1.4

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

$(- \frac{21}{y^{4}} + 36 \frac{1}{y^{5}}) (y^{- 1} + y^{- 2}) + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.1.5

Combine $36$ and $\frac{1}{y^{5}}$ .

$(- \frac{21}{y^{4}} + \frac{36}{y^{5}}) (y^{- 1} + y^{- 2}) + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

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

$(- \frac{21}{y^{4}} + \frac{36}{y^{5}}) (\frac{1}{y} + y^{- 2}) + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.2.2

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

$(- \frac{21}{y^{4}} + \frac{36}{y^{5}}) (\frac{1}{y} + \frac{1}{y^{2}}) + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.3

Expand $(- \frac{21}{y^{4}} + \frac{36}{y^{5}}) (\frac{1}{y} + \frac{1}{y^{2}})$ using the FOIL Method.

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Step 3.2.3.1

Apply the distributive property.

$- \frac{21}{y^{4}} (\frac{1}{y} + \frac{1}{y^{2}}) + \frac{36}{y^{5}} (\frac{1}{y} + \frac{1}{y^{2}}) + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.3.2

Apply the distributive property.

$- \frac{21}{y^{4}} \cdot \frac{1}{y} - \frac{21}{y^{4}} \cdot \frac{1}{y^{2}} + \frac{36}{y^{5}} (\frac{1}{y} + \frac{1}{y^{2}}) + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.3.3

Apply the distributive property.

$- \frac{21}{y^{4}} \cdot \frac{1}{y} - \frac{21}{y^{4}} \cdot \frac{1}{y^{2}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4

Simplify and combine like terms.

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

Multiply $- \frac{21}{y^{4}} \cdot \frac{1}{y}$ .

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Step 3.2.4.1.1.1

Multiply $\frac{1}{y}$ by $\frac{21}{y^{4}}$ .

$- \frac{21}{y \cdot y^{4}} - \frac{21}{y^{4}} \cdot \frac{1}{y^{2}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.1.2

Multiply $y$ by $y^{4}$ by adding the exponents.

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Step 3.2.4.1.1.2.1

Multiply $y$ by $y^{4}$ .

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Step 3.2.4.1.1.2.1.1

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

$- \frac{21}{y^{1} y^{4}} - \frac{21}{y^{4}} \cdot \frac{1}{y^{2}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.1.2.1.2

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

$- \frac{21}{y^{1 + 4}} - \frac{21}{y^{4}} \cdot \frac{1}{y^{2}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.1.2.2

Add $1$ and $4$ .

$- \frac{21}{y^{5}} - \frac{21}{y^{4}} \cdot \frac{1}{y^{2}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.2

Multiply $- \frac{21}{y^{4}} \cdot \frac{1}{y^{2}}$ .

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Step 3.2.4.1.2.1

Multiply $\frac{1}{y^{2}}$ by $\frac{21}{y^{4}}$ .

$- \frac{21}{y^{5}} - \frac{21}{y^{2} y^{4}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.2.2

Multiply $y^{2}$ by $y^{4}$ by adding the exponents.

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Step 3.2.4.1.2.2.1

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

$- \frac{21}{y^{5}} - \frac{21}{y^{2 + 4}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.2.2.2

Add $2$ and $4$ .

$- \frac{21}{y^{5}} - \frac{21}{y^{6}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.3

Combine.

$- \frac{21}{y^{5}} - \frac{21}{y^{6}} + \frac{36 \cdot 1}{y^{5} y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.4

Multiply $y^{5}$ by $y$ by adding the exponents.

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Step 3.2.4.1.4.1

Multiply $y^{5}$ by $y$ .

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Step 3.2.4.1.4.1.1

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

$- \frac{21}{y^{5}} - \frac{21}{y^{6}} + \frac{36 \cdot 1}{y^{5} y^{1}} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.4.1.2

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

$- \frac{21}{y^{5}} - \frac{21}{y^{6}} + \frac{36 \cdot 1}{y^{5 + 1}} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.4.2

Add $5$ and $1$ .

$- \frac{21}{y^{5}} - \frac{21}{y^{6}} + \frac{36 \cdot 1}{y^{6}} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.5

Multiply $36$ by $1$ .

$- \frac{21}{y^{5}} - \frac{21}{y^{6}} + \frac{36}{y^{6}} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.6

Combine.

$- \frac{21}{y^{5}} - \frac{21}{y^{6}} + \frac{36}{y^{6}} + \frac{36 \cdot 1}{y^{5} y^{2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.7

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

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Step 3.2.4.1.7.1

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

$- \frac{21}{y^{5}} - \frac{21}{y^{6}} + \frac{36}{y^{6}} + \frac{36 \cdot 1}{y^{5 + 2}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.7.2

Add $5$ and $2$ .

$- \frac{21}{y^{5}} - \frac{21}{y^{6}} + \frac{36}{y^{6}} + \frac{36 \cdot 1}{y^{7}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.1.8

Multiply $36$ by $1$ .

$- \frac{21}{y^{5}} - \frac{21}{y^{6}} + \frac{36}{y^{6}} + \frac{36}{y^{7}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.2

Combine the numerators over the common denominator.

$- \frac{21}{y^{5}} + \frac{- 21 + 36}{y^{6}} + \frac{36}{y^{7}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.4.3

Add $- 21$ and $36$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} + (- y^{- 2} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.5

Simplify each term.

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Step 3.2.5.1

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

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - 2 y^{- 3}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.5.2

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

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - 2 \frac{1}{y^{3}}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.5.3

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

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} + \frac{- 2}{y^{3}}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.5.4

Move the negative in front of the fraction.

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - \frac{2}{y^{3}}) (7 y^{- 3} - 9 y^{- 4})$

Step 3.2.6

Simplify each term.

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Step 3.2.6.1

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

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - \frac{2}{y^{3}}) (7 \frac{1}{y^{3}} - 9 y^{- 4})$

Step 3.2.6.2

Combine $7$ and $\frac{1}{y^{3}}$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - \frac{2}{y^{3}}) (\frac{7}{y^{3}} - 9 y^{- 4})$

Step 3.2.6.3

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

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - \frac{2}{y^{3}}) (\frac{7}{y^{3}} - 9 \frac{1}{y^{4}})$

Step 3.2.6.4

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

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - \frac{2}{y^{3}}) (\frac{7}{y^{3}} + \frac{- 9}{y^{4}})$

Step 3.2.6.5

Move the negative in front of the fraction.

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - \frac{2}{y^{3}}) (\frac{7}{y^{3}} - \frac{9}{y^{4}})$

Step 3.2.7

Expand $(- \frac{1}{y^{2}} - \frac{2}{y^{3}}) (\frac{7}{y^{3}} - \frac{9}{y^{4}})$ using the FOIL Method.

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Step 3.2.7.1

Apply the distributive property.

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{1}{y^{2}} (\frac{7}{y^{3}} - \frac{9}{y^{4}}) - \frac{2}{y^{3}} (\frac{7}{y^{3}} - \frac{9}{y^{4}})$

Step 3.2.7.2

Apply the distributive property.

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{1}{y^{2}} \cdot \frac{7}{y^{3}} - \frac{1}{y^{2}} (- \frac{9}{y^{4}}) - \frac{2}{y^{3}} (\frac{7}{y^{3}} - \frac{9}{y^{4}})$

Step 3.2.7.3

Apply the distributive property.

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{1}{y^{2}} \cdot \frac{7}{y^{3}} - \frac{1}{y^{2}} (- \frac{9}{y^{4}}) - \frac{2}{y^{3}} \cdot \frac{7}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Step 3.2.8

Simplify and combine like terms.

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Step 3.2.8.1

Simplify each term.

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Step 3.2.8.1.1

Multiply $- \frac{1}{y^{2}} \cdot \frac{7}{y^{3}}$ .

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Step 3.2.8.1.1.1

Multiply $\frac{7}{y^{3}}$ by $\frac{1}{y^{2}}$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{3} y^{2}} - \frac{1}{y^{2}} (- \frac{9}{y^{4}}) - \frac{2}{y^{3}} \cdot \frac{7}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Step 3.2.8.1.1.2

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

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Step 3.2.8.1.1.2.1

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

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{3 + 2}} - \frac{1}{y^{2}} (- \frac{9}{y^{4}}) - \frac{2}{y^{3}} \cdot \frac{7}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Step 3.2.8.1.1.2.2

Add $3$ and $2$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} - \frac{1}{y^{2}} (- \frac{9}{y^{4}}) - \frac{2}{y^{3}} \cdot \frac{7}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Step 3.2.8.1.2

Multiply $- \frac{1}{y^{2}} (- \frac{9}{y^{4}})$ .

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Step 3.2.8.1.2.1

Multiply $- 1$ by $- 1$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + 1 \frac{1}{y^{2}} \frac{9}{y^{4}} - \frac{2}{y^{3}} \cdot \frac{7}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Step 3.2.8.1.2.2

Multiply $\frac{1}{y^{2}}$ by $1$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{1}{y^{2}} \cdot \frac{9}{y^{4}} - \frac{2}{y^{3}} \cdot \frac{7}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Step 3.2.8.1.2.3

Multiply $\frac{1}{y^{2}}$ by $\frac{9}{y^{4}}$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{9}{y^{2} y^{4}} - \frac{2}{y^{3}} \cdot \frac{7}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Step 3.2.8.1.2.4

Multiply $y^{2}$ by $y^{4}$ by adding the exponents.

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Step 3.2.8.1.2.4.1

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

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{9}{y^{2 + 4}} - \frac{2}{y^{3}} \cdot \frac{7}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Step 3.2.8.1.2.4.2

Add $2$ and $4$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{9}{y^{6}} - \frac{2}{y^{3}} \cdot \frac{7}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Step 3.2.8.1.3

Multiply $- \frac{2}{y^{3}} \cdot \frac{7}{y^{3}}$ .

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Step 3.2.8.1.3.1

Multiply $\frac{7}{y^{3}}$ by $\frac{2}{y^{3}}$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{9}{y^{6}} - \frac{7 \cdot 2}{y^{3} y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Step 3.2.8.1.3.2

Multiply $7$ by $2$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{9}{y^{6}} - \frac{14}{y^{3} y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Step 3.2.8.1.3.3

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

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Step 3.2.8.1.3.3.1

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

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{9}{y^{6}} - \frac{14}{y^{3 + 3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Step 3.2.8.1.3.3.2

Add $3$ and $3$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{9}{y^{6}} - \frac{14}{y^{6}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Step 3.2.8.1.4

Multiply $- \frac{2}{y^{3}} (- \frac{9}{y^{4}})$ .

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Step 3.2.8.1.4.1

Multiply $- 1$ by $- 1$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{9}{y^{6}} - \frac{14}{y^{6}} + 1 \frac{2}{y^{3}} \frac{9}{y^{4}}$

Step 3.2.8.1.4.2

Multiply $\frac{2}{y^{3}}$ by $1$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{9}{y^{6}} - \frac{14}{y^{6}} + \frac{2}{y^{3}} \cdot \frac{9}{y^{4}}$

Step 3.2.8.1.4.3

Multiply $\frac{2}{y^{3}}$ by $\frac{9}{y^{4}}$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{9}{y^{6}} - \frac{14}{y^{6}} + \frac{2 \cdot 9}{y^{3} y^{4}}$

Step 3.2.8.1.4.4

Multiply $2$ by $9$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{9}{y^{6}} - \frac{14}{y^{6}} + \frac{18}{y^{3} y^{4}}$

Step 3.2.8.1.4.5

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

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Step 3.2.8.1.4.5.1

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

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{9}{y^{6}} - \frac{14}{y^{6}} + \frac{18}{y^{3 + 4}}$

Step 3.2.8.1.4.5.2

Add $3$ and $4$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{9}{y^{6}} - \frac{14}{y^{6}} + \frac{18}{y^{7}}$

Step 3.2.8.2

Combine the numerators over the common denominator.

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{9 - 14}{y^{6}} + \frac{18}{y^{7}}$

Step 3.2.8.3

Subtract $14$ from $9$ .

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} + \frac{- 5}{y^{6}} + \frac{18}{y^{7}}$

Step 3.2.9

Move the negative in front of the fraction.

$- \frac{21}{y^{5}} + \frac{15}{y^{6}} + \frac{36}{y^{7}} - \frac{7}{y^{5}} - \frac{5}{y^{6}} + \frac{18}{y^{7}}$

Step 3.3

Combine the numerators over the common denominator.

$\frac{- 21 - 7}{y^{5}} + \frac{15 - 5}{y^{6}} + \frac{36 + 18}{y^{7}}$

Step 3.4

Subtract $7$ from $- 21$ .

$\frac{- 28}{y^{5}} + \frac{15 - 5}{y^{6}} + \frac{36 + 18}{y^{7}}$

Step 3.5

Subtract $5$ from $15$ .

$\frac{- 28}{y^{5}} + \frac{10}{y^{6}} + \frac{36 + 18}{y^{7}}$

Step 3.6

Add $36$ and $18$ .

$\frac{- 28}{y^{5}} + \frac{10}{y^{6}} + \frac{54}{y^{7}}$

Step 3.7

Move the negative in front of the fraction.

$- \frac{28}{y^{5}} + \frac{10}{y^{6}} + \frac{54}{y^{7}}$