Calculus Examples

$F (y) = \frac{1}{y^{2}} - \frac{5}{y^{4}} \cdot (y + 9 y^{3})$

Step 1

By the Sum Rule, the derivative of $\frac{1}{y^{2}} - \frac{5}{y^{4}} \cdot (y + 9 y^{3})$ with respect to $y$ is $\frac{d}{d y} [\frac{1}{y^{2}}] + \frac{d}{d y} [- \frac{5}{y^{4}} \cdot (y + 9 y^{3})]$ .

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

Step 2

Evaluate $\frac{d}{d y} [\frac{1}{y^{2}}]$ .

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Step 2.1

Rewrite $\frac{1}{y^{2}}$ as ${(y^{2})}^{- 1}$ .

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

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{- 1}$ and $g (y) = y^{2}$ .

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Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $y^{2}$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u_{1}$ with $y^{2}$ .

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

Step 2.3

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

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

Step 2.4

Multiply the exponents in ${(y^{2})}^{- 2}$ .

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Step 2.4.1

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

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

Step 2.4.2

Multiply $2$ by $- 2$ .

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

Step 2.5

Multiply $2$ by $- 1$ .

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Subtract $4$ from $1$ .

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

Step 3

Evaluate $\frac{d}{d y} [- \frac{5}{y^{4}} \cdot (y + 9 y^{3})]$ .

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Step 3.1

Since $- 1$ is constant with respect to $y$ , the derivative of $- \frac{5}{y^{4}} (y + 9 y^{3})$ with respect to $y$ is $- \frac{d}{d y} [\frac{5}{y^{4}} (y + 9 y^{3})]$ .

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

Step 3.2

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 $f (y) = \frac{5}{y^{4}}$ and $g (y) = y + 9 y^{3}$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Since $9$ is constant with respect to $y$ , the derivative of $9 y^{3}$ with respect to $y$ is $9 \frac{d}{d y} [y^{3}]$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{- 1}$ and $g (y) = y^{4}$ .

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Step 3.9.1

To apply the Chain Rule, set $u_{2}$ as $y^{4}$ .

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

Step 3.9.2

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

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

Step 3.9.3

Replace all occurrences of $u_{2}$ with $y^{4}$ .

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

Step 3.10

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

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

Step 3.11

Multiply $3$ by $9$ .

$- 2 y^{- 3} - (\frac{5}{y^{4}} (1 + 27 y^{2}) + (y + 9 y^{3}) (5 (- {(y^{4})}^{- 2} (4 y^{3}))))$

Step 3.12

Multiply the exponents in ${(y^{4})}^{- 2}$ .

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Step 3.12.1

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

$- 2 y^{- 3} - (\frac{5}{y^{4}} (1 + 27 y^{2}) + (y + 9 y^{3}) (5 (- y^{4 \cdot - 2} (4 y^{3}))))$

Step 3.12.2

Multiply $4$ by $- 2$ .

$- 2 y^{- 3} - (\frac{5}{y^{4}} (1 + 27 y^{2}) + (y + 9 y^{3}) (5 (- y^{- 8} (4 y^{3}))))$

Step 3.13

Multiply $4$ by $- 1$ .

$- 2 y^{- 3} - (\frac{5}{y^{4}} (1 + 27 y^{2}) + (y + 9 y^{3}) (5 (- 4 y^{- 8} y^{3})))$

Step 3.14

Multiply $y^{- 8}$ by $y^{3}$ by adding the exponents.

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Step 3.14.1

Move $y^{3}$ .

$- 2 y^{- 3} - (\frac{5}{y^{4}} (1 + 27 y^{2}) + (y + 9 y^{3}) (5 (- 4 (y^{3} y^{- 8}))))$

Step 3.14.2

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

$- 2 y^{- 3} - (\frac{5}{y^{4}} (1 + 27 y^{2}) + (y + 9 y^{3}) (5 (- 4 y^{3 - 8})))$

Step 3.14.3

Subtract $8$ from $3$ .

$- 2 y^{- 3} - (\frac{5}{y^{4}} (1 + 27 y^{2}) + (y + 9 y^{3}) (5 (- 4 y^{- 5})))$

Step 3.15

Multiply $- 4$ by $5$ .

$- 2 y^{- 3} - (\frac{5}{y^{4}} (1 + 27 y^{2}) + (y + 9 y^{3}) (- 20 y^{- 5}))$

Step 4

Simplify.

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Step 4.1

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

$- 2 \frac{1}{y^{3}} - (\frac{5}{y^{4}} (1 + 27 y^{2}) + (y + 9 y^{3}) (- 20 y^{- 5}))$

Step 4.2

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

$- 2 \frac{1}{y^{3}} - (\frac{5}{y^{4}} (1 + 27 y^{2}) + (y + 9 y^{3}) (- 20 \frac{1}{y^{5}}))$

Step 4.3

Apply the distributive property.

$- 2 \frac{1}{y^{3}} - (\frac{5}{y^{4}} \cdot 1 + \frac{5}{y^{4}} (27 y^{2}) + (y + 9 y^{3}) (- 20 \frac{1}{y^{5}}))$

Step 4.4

Apply the distributive property.

$- 2 \frac{1}{y^{3}} - (\frac{5}{y^{4}} \cdot 1 + \frac{5}{y^{4}} (27 y^{2}) + y (- 20 \frac{1}{y^{5}}) + 9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.5

Apply the distributive property.

$- 2 \frac{1}{y^{3}} - (\frac{5}{y^{4}} \cdot 1) - (\frac{5}{y^{4}} (27 y^{2})) - (y (- 20 \frac{1}{y^{5}})) - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6

Combine terms.

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Step 4.6.1

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

$\frac{- 2}{y^{3}} - (\frac{5}{y^{4}} \cdot 1) - (\frac{5}{y^{4}} (27 y^{2})) - (y (- 20 \frac{1}{y^{5}})) - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.2

Move the negative in front of the fraction.

$- \frac{2}{y^{3}} - (\frac{5}{y^{4}} \cdot 1) - (\frac{5}{y^{4}} (27 y^{2})) - (y (- 20 \frac{1}{y^{5}})) - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.3

Multiply $\frac{5}{y^{4}}$ by $1$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - (\frac{5}{y^{4}} (27 y^{2})) - (y (- 20 \frac{1}{y^{5}})) - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.4

Combine $27$ and $\frac{5}{y^{4}}$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - (\frac{27 \cdot 5}{y^{4}} y^{2}) - (y (- 20 \frac{1}{y^{5}})) - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.5

Multiply $27$ by $5$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - (\frac{135}{y^{4}} y^{2}) - (y (- 20 \frac{1}{y^{5}})) - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.6

Combine $\frac{135}{y^{4}}$ and $y^{2}$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135 y^{2}}{y^{4}} - (y (- 20 \frac{1}{y^{5}})) - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.7

Cancel the common factor of $y^{2}$ and $y^{4}$ .

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Step 4.6.7.1

Factor $y^{2}$ out of $135 y^{2}$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{y^{2} \cdot 135}{y^{4}} - (y (- 20 \frac{1}{y^{5}})) - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.7.2

Cancel the common factors.

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Step 4.6.7.2.1

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

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{y^{2} \cdot 135}{y^{2} y^{2}} - (y (- 20 \frac{1}{y^{5}})) - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.7.2.2

Cancel the common factor.

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{y^{2} \cdot 135}{y^{2} y^{2}} - (y (- 20 \frac{1}{y^{5}})) - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.7.2.3

Rewrite the expression.

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} - (y (- 20 \frac{1}{y^{5}})) - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.8

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

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} - (y \frac{- 20}{y^{5}}) - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.9

Move the negative in front of the fraction.

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} - (y (- \frac{20}{y^{5}})) - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.10

Combine $y$ and $\frac{20}{y^{5}}$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} - - \frac{y \cdot 20}{y^{5}} - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.11

Move $20$ to the left of $y$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} - - \frac{20 \cdot y}{y^{5}} - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.12

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

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Step 4.6.12.1

Factor $y$ out of $20 y$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} - - \frac{y \cdot 20}{y^{5}} - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.12.2

Cancel the common factors.

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Step 4.6.12.2.1

Factor $y$ out of $y^{5}$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} - - \frac{y \cdot 20}{y \cdot y^{4}} - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.12.2.2

Cancel the common factor.

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} - - \frac{y \cdot 20}{y \cdot y^{4}} - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.12.2.3

Rewrite the expression.

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} - - \frac{20}{y^{4}} - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.13

Multiply $- 1$ by $- 1$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + 1 \frac{20}{y^{4}} - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.14

Multiply $\frac{20}{y^{4}}$ by $1$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} - (9 y^{3} (- 20 \frac{1}{y^{5}}))$

Step 4.6.15

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

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} - (9 y^{3} \frac{- 20}{y^{5}})$

Step 4.6.16

Move the negative in front of the fraction.

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} - (9 y^{3} (- \frac{20}{y^{5}}))$

Step 4.6.17

Multiply $- 1$ by $9$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} - (- 9 y^{3} \frac{20}{y^{5}})$

Step 4.6.18

Combine $- 9$ and $\frac{20}{y^{5}}$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} - (y^{3} \frac{- 9 \cdot 20}{y^{5}})$

Step 4.6.19

Multiply $- 9$ by $20$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} - (y^{3} \frac{- 180}{y^{5}})$

Step 4.6.20

Combine $y^{3}$ and $\frac{- 180}{y^{5}}$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} - \frac{y^{3} \cdot - 180}{y^{5}}$

Step 4.6.21

Move $- 180$ to the left of $y^{3}$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} - \frac{- 180 \cdot y^{3}}{y^{5}}$

Step 4.6.22

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

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Step 4.6.22.1

Factor $y^{3}$ out of $- 180 y^{3}$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} - \frac{y^{3} \cdot - 180}{y^{5}}$

Step 4.6.22.2

Cancel the common factors.

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Step 4.6.22.2.1

Factor $y^{3}$ out of $y^{5}$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} - \frac{y^{3} \cdot - 180}{y^{3} y^{2}}$

Step 4.6.22.2.2

Cancel the common factor.

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} - \frac{y^{3} \cdot - 180}{y^{3} y^{2}}$

Step 4.6.22.2.3

Rewrite the expression.

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} - \frac{- 180}{y^{2}}$

Step 4.6.23

Move the negative in front of the fraction.

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} - - \frac{180}{y^{2}}$

Step 4.6.24

Multiply $- 1$ by $- 1$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} + 1 \frac{180}{y^{2}}$

Step 4.6.25

Multiply $\frac{180}{y^{2}}$ by $1$ .

$- \frac{2}{y^{3}} - \frac{5}{y^{4}} - \frac{135}{y^{2}} + \frac{20}{y^{4}} + \frac{180}{y^{2}}$

Step 4.6.26

Combine the numerators over the common denominator.

$- \frac{2}{y^{3}} - \frac{135}{y^{2}} + \frac{- 5 + 20}{y^{4}} + \frac{180}{y^{2}}$

Step 4.6.27

Add $- 5$ and $20$ .

$- \frac{2}{y^{3}} - \frac{135}{y^{2}} + \frac{15}{y^{4}} + \frac{180}{y^{2}}$

Step 4.6.28

Combine the numerators over the common denominator.

$- \frac{2}{y^{3}} + \frac{- 135 + 180}{y^{2}} + \frac{15}{y^{4}}$

Step 4.6.29

Add $- 135$ and $180$ .

$- \frac{2}{y^{3}} + \frac{45}{y^{2}} + \frac{15}{y^{4}}$

Step 4.7

Reorder terms.

$\frac{45}{y^{2}} - \frac{2}{y^{3}} + \frac{15}{y^{4}}$