Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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 $f (y) = \frac{1}{y^{2}} - \frac{5}{y^{4}}$ and $g (y) = y + 3 y^{3}$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Multiply $3$ by $3$ .

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

Step 1.2.6

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

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

Step 1.2.7

Apply basic rules of exponents.

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Step 1.2.7.1

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

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

Step 1.2.7.2

Multiply the exponents in ${(y^{2})}^{- 1}$ .

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Step 1.2.7.2.1

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

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

Step 1.2.7.2.2

Multiply $2$ by $- 1$ .

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

Step 1.2.8

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

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

Step 1.2.9

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

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

Step 1.2.10

Apply basic rules of exponents.

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Step 1.2.10.1

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

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

Step 1.2.10.2

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

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Step 1.2.10.2.1

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

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

Step 1.2.10.2.2

Multiply $4$ by $- 1$ .

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

Step 1.2.11

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

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

Step 1.2.12

Multiply $- 4$ by $- 5$ .

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

Step 1.3

Simplify.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Combine terms.

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Step 1.3.3.1

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

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

Step 1.3.3.2

Move the negative in front of the fraction.

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

Step 1.3.3.3

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

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

Step 1.3.4

Reorder terms.

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

Step 1.3.5

Simplify each term.

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Step 1.3.5.1

Expand $(1 + 9 y^{2}) (\frac{1}{y^{2}} - \frac{5}{y^{4}})$ using the FOIL Method.

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Step 1.3.5.1.1

Apply the distributive property.

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

Step 1.3.5.1.2

Apply the distributive property.

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

Step 1.3.5.1.3

Apply the distributive property.

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

Step 1.3.5.2

Simplify and combine like terms.

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Step 1.3.5.2.1

Simplify each term.

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Step 1.3.5.2.1.1

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

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

Step 1.3.5.2.1.2

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

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

Step 1.3.5.2.1.3

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

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Step 1.3.5.2.1.3.1

Factor $y^{2}$ out of $9 y^{2}$ .

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

Step 1.3.5.2.1.3.2

Cancel the common factor.

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

Step 1.3.5.2.1.3.3

Rewrite the expression.

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

Step 1.3.5.2.1.4

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

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Step 1.3.5.2.1.4.1

Move the leading negative in $- \frac{5}{y^{4}}$ into the numerator.

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

Step 1.3.5.2.1.4.2

Factor $y^{2}$ out of $9 y^{2}$ .

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

Step 1.3.5.2.1.4.3

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

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

Step 1.3.5.2.1.4.4

Cancel the common factor.

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

Step 1.3.5.2.1.4.5

Rewrite the expression.

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

Step 1.3.5.2.1.5

Combine $9$ and $\frac{- 5}{y^{2}}$ .

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

Step 1.3.5.2.1.6

Multiply $9$ by $- 5$ .

$\frac{1}{y^{2}} - \frac{5}{y^{4}} + 9 + \frac{- 45}{y^{2}} + (- \frac{2}{y^{3}} + \frac{20}{y^{5}}) (y + 3 y^{3})$

Step 1.3.5.2.1.7

Move the negative in front of the fraction.

$\frac{1}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{45}{y^{2}} + (- \frac{2}{y^{3}} + \frac{20}{y^{5}}) (y + 3 y^{3})$

Step 1.3.5.2.2

Combine the numerators over the common denominator.

$\frac{1 - 45}{y^{2}} - \frac{5}{y^{4}} + 9 + (- \frac{2}{y^{3}} + \frac{20}{y^{5}}) (y + 3 y^{3})$

Step 1.3.5.2.3

Subtract $45$ from $1$ .

$\frac{- 44}{y^{2}} - \frac{5}{y^{4}} + 9 + (- \frac{2}{y^{3}} + \frac{20}{y^{5}}) (y + 3 y^{3})$

Step 1.3.5.3

Move the negative in front of the fraction.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 + (- \frac{2}{y^{3}} + \frac{20}{y^{5}}) (y + 3 y^{3})$

Step 1.3.5.4

Expand $(- \frac{2}{y^{3}} + \frac{20}{y^{5}}) (y + 3 y^{3})$ using the FOIL Method.

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Step 1.3.5.4.1

Apply the distributive property.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{3}} (y + 3 y^{3}) + \frac{20}{y^{5}} (y + 3 y^{3})$

Step 1.3.5.4.2

Apply the distributive property.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{3}} y - \frac{2}{y^{3}} (3 y^{3}) + \frac{20}{y^{5}} (y + 3 y^{3})$

Step 1.3.5.4.3

Apply the distributive property.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{3}} y - \frac{2}{y^{3}} (3 y^{3}) + \frac{20}{y^{5}} y + \frac{20}{y^{5}} (3 y^{3})$

Step 1.3.5.5

Simplify and combine like terms.

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Step 1.3.5.5.1

Simplify each term.

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Step 1.3.5.5.1.1

Cancel the common factor of $y$ .

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Step 1.3.5.5.1.1.1

Move the leading negative in $- \frac{2}{y^{3}}$ into the numerator.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 + \frac{- 2}{y^{3}} y - \frac{2}{y^{3}} (3 y^{3}) + \frac{20}{y^{5}} y + \frac{20}{y^{5}} (3 y^{3})$

Step 1.3.5.5.1.1.2

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

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 + \frac{- 2}{y \cdot y^{2}} y - \frac{2}{y^{3}} (3 y^{3}) + \frac{20}{y^{5}} y + \frac{20}{y^{5}} (3 y^{3})$

Step 1.3.5.5.1.1.3

Cancel the common factor.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 + \frac{- 2}{y \cdot y^{2}} y - \frac{2}{y^{3}} (3 y^{3}) + \frac{20}{y^{5}} y + \frac{20}{y^{5}} (3 y^{3})$

Step 1.3.5.5.1.1.4

Rewrite the expression.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 + \frac{- 2}{y^{2}} - \frac{2}{y^{3}} (3 y^{3}) + \frac{20}{y^{5}} y + \frac{20}{y^{5}} (3 y^{3})$

Step 1.3.5.5.1.2

Move the negative in front of the fraction.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} - \frac{2}{y^{3}} (3 y^{3}) + \frac{20}{y^{5}} y + \frac{20}{y^{5}} (3 y^{3})$

Step 1.3.5.5.1.3

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

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Step 1.3.5.5.1.3.1

Move the leading negative in $- \frac{2}{y^{3}}$ into the numerator.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} + \frac{- 2}{y^{3}} (3 y^{3}) + \frac{20}{y^{5}} y + \frac{20}{y^{5}} (3 y^{3})$

Step 1.3.5.5.1.3.2

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

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} + \frac{- 2}{y^{3}} (y^{3} \cdot 3) + \frac{20}{y^{5}} y + \frac{20}{y^{5}} (3 y^{3})$

Step 1.3.5.5.1.3.3

Cancel the common factor.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} + \frac{- 2}{y^{3}} (y^{3} \cdot 3) + \frac{20}{y^{5}} y + \frac{20}{y^{5}} (3 y^{3})$

Step 1.3.5.5.1.3.4

Rewrite the expression.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} - 2 \cdot 3 + \frac{20}{y^{5}} y + \frac{20}{y^{5}} (3 y^{3})$

Step 1.3.5.5.1.4

Multiply $- 2$ by $3$ .

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} - 6 + \frac{20}{y^{5}} y + \frac{20}{y^{5}} (3 y^{3})$

Step 1.3.5.5.1.5

Cancel the common factor of $y$ .

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Step 1.3.5.5.1.5.1

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

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} - 6 + \frac{20}{y \cdot y^{4}} y + \frac{20}{y^{5}} (3 y^{3})$

Step 1.3.5.5.1.5.2

Cancel the common factor.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} - 6 + \frac{20}{y \cdot y^{4}} y + \frac{20}{y^{5}} (3 y^{3})$

Step 1.3.5.5.1.5.3

Rewrite the expression.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} - 6 + \frac{20}{y^{4}} + \frac{20}{y^{5}} (3 y^{3})$

Step 1.3.5.5.1.6

Rewrite using the commutative property of multiplication.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} - 6 + \frac{20}{y^{4}} + 3 \frac{20}{y^{5}} y^{3}$

Step 1.3.5.5.1.7

Multiply $3 \frac{20}{y^{5}}$ .

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Step 1.3.5.5.1.7.1

Combine $3$ and $\frac{20}{y^{5}}$ .

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} - 6 + \frac{20}{y^{4}} + \frac{3 \cdot 20}{y^{5}} y^{3}$

Step 1.3.5.5.1.7.2

Multiply $3$ by $20$ .

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} - 6 + \frac{20}{y^{4}} + \frac{60}{y^{5}} y^{3}$

Step 1.3.5.5.1.8

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

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Step 1.3.5.5.1.8.1

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

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} - 6 + \frac{20}{y^{4}} + \frac{60}{y^{3} y^{2}} y^{3}$

Step 1.3.5.5.1.8.2

Cancel the common factor.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} - 6 + \frac{20}{y^{4}} + \frac{60}{y^{3} y^{2}} y^{3}$

Step 1.3.5.5.1.8.3

Rewrite the expression.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 - \frac{2}{y^{2}} - 6 + \frac{20}{y^{4}} + \frac{60}{y^{2}}$

Step 1.3.5.5.2

Combine the numerators over the common denominator.

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 + \frac{- 2 + 60}{y^{2}} - 6 + \frac{20}{y^{4}}$

Step 1.3.5.5.3

Add $- 2$ and $60$ .

$- \frac{44}{y^{2}} - \frac{5}{y^{4}} + 9 + \frac{58}{y^{2}} - 6 + \frac{20}{y^{4}}$

Step 1.3.6

Combine the numerators over the common denominator.

$9 - 6 + \frac{- 44 + 58}{y^{2}} + \frac{- 5 + 20}{y^{4}}$

Step 1.3.7

Add $- 44$ and $58$ .

$9 - 6 + \frac{14}{y^{2}} + \frac{- 5 + 20}{y^{4}}$

Step 1.3.8

Add $- 5$ and $20$ .

$9 - 6 + \frac{14}{y^{2}} + \frac{15}{y^{4}}$

Step 1.3.9

Subtract $6$ from $9$ .

$f' (y) = 3 + \frac{14}{y^{2}} + \frac{15}{y^{4}}$

Step 2

Find the second derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $3 + \frac{14}{y^{2}} + \frac{15}{y^{4}}$ with respect to $y$ is $\frac{d}{d y} [3] + \frac{d}{d y} [\frac{14}{y^{2}}] + \frac{d}{d y} [\frac{15}{y^{4}}]$ .

$\frac{d}{d y} [3] + \frac{d}{d y} [\frac{14}{y^{2}}] + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.1.2

Since $3$ is constant with respect to $y$ , the derivative of $3$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [\frac{14}{y^{2}}] + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.2

Evaluate $\frac{d}{d y} [\frac{14}{y^{2}}]$ .

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Step 2.2.1

Since $14$ is constant with respect to $y$ , the derivative of $\frac{14}{y^{2}}$ with respect to $y$ is $14 \frac{d}{d y} [\frac{1}{y^{2}}]$ .

$0 + 14 \frac{d}{d y} [\frac{1}{y^{2}}] + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.2.2

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

$0 + 14 \frac{d}{d y} [{(y^{2})}^{- 1}] + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.2.3

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 2.2.3.1

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

$0 + 14 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d y} [y^{2}]) + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.2.3.2

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

$0 + 14 (- {u_{1}}^{- 2} \frac{d}{d y} [y^{2}]) + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.2.3.3

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

$0 + 14 (- {(y^{2})}^{- 2} \frac{d}{d y} [y^{2}]) + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.2.4

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

$0 + 14 (- {(y^{2})}^{- 2} (2 y)) + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.2.5

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

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Step 2.2.5.1

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

$0 + 14 (- y^{2 \cdot - 2} (2 y)) + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.2.5.2

Multiply $2$ by $- 2$ .

$0 + 14 (- y^{- 4} (2 y)) + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.2.6

Multiply $2$ by $- 1$ .

$0 + 14 (- 2 y^{- 4} y) + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.2.7

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

$0 + 14 (- 2 (y^{1} y^{- 4})) + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.2.8

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

$0 + 14 (- 2 y^{1 - 4}) + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.2.9

Subtract $4$ from $1$ .

$0 + 14 (- 2 y^{- 3}) + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.2.10

Multiply $- 2$ by $14$ .

$0 - 28 y^{- 3} + \frac{d}{d y} [\frac{15}{y^{4}}]$

Step 2.3

Evaluate $\frac{d}{d y} [\frac{15}{y^{4}}]$ .

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Step 2.3.1

Since $15$ is constant with respect to $y$ , the derivative of $\frac{15}{y^{4}}$ with respect to $y$ is $15 \frac{d}{d y} [\frac{1}{y^{4}}]$ .

$0 - 28 y^{- 3} + 15 \frac{d}{d y} [\frac{1}{y^{4}}]$

Step 2.3.2

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

$0 - 28 y^{- 3} + 15 \frac{d}{d y} [{(y^{4})}^{- 1}]$

Step 2.3.3

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 2.3.3.1

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

$0 - 28 y^{- 3} + 15 (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d y} [y^{4}])$

Step 2.3.3.2

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

$0 - 28 y^{- 3} + 15 (- {u_{2}}^{- 2} \frac{d}{d y} [y^{4}])$

Step 2.3.3.3

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

$0 - 28 y^{- 3} + 15 (- {(y^{4})}^{- 2} \frac{d}{d y} [y^{4}])$

Step 2.3.4

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

$0 - 28 y^{- 3} + 15 (- {(y^{4})}^{- 2} (4 y^{3}))$

Step 2.3.5

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

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Step 2.3.5.1

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

$0 - 28 y^{- 3} + 15 (- y^{4 \cdot - 2} (4 y^{3}))$

Step 2.3.5.2

Multiply $4$ by $- 2$ .

$0 - 28 y^{- 3} + 15 (- y^{- 8} (4 y^{3}))$

Step 2.3.6

Multiply $4$ by $- 1$ .

$0 - 28 y^{- 3} + 15 (- 4 y^{- 8} y^{3})$

Step 2.3.7

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

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Step 2.3.7.1

Move $y^{3}$ .

$0 - 28 y^{- 3} + 15 (- 4 (y^{3} y^{- 8}))$

Step 2.3.7.2

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

$0 - 28 y^{- 3} + 15 (- 4 y^{3 - 8})$

Step 2.3.7.3

Subtract $8$ from $3$ .

$0 - 28 y^{- 3} + 15 (- 4 y^{- 5})$

Step 2.3.8

Multiply $- 4$ by $15$ .

$0 - 28 y^{- 3} - 60 y^{- 5}$

Step 2.4

Simplify.

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Step 2.4.1

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

$0 - 28 \frac{1}{y^{3}} - 60 y^{- 5}$

Step 2.4.2

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

$0 - 28 \frac{1}{y^{3}} - 60 \frac{1}{y^{5}}$

Step 2.4.3

Combine terms.

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Step 2.4.3.1

Combine $- 28$ and $\frac{1}{y^{3}}$ .

$0 + \frac{- 28}{y^{3}} - 60 \frac{1}{y^{5}}$

Step 2.4.3.2

Move the negative in front of the fraction.

$0 - \frac{28}{y^{3}} - 60 \frac{1}{y^{5}}$

Step 2.4.3.3

Subtract $\frac{28}{y^{3}}$ from $0$ .

$- \frac{28}{y^{3}} - 60 \frac{1}{y^{5}}$

Step 2.4.3.4

Combine $- 60$ and $\frac{1}{y^{5}}$ .

$- \frac{28}{y^{3}} + \frac{- 60}{y^{5}}$

Step 2.4.3.5

Move the negative in front of the fraction.

$f'' (y) = - \frac{28}{y^{3}} - \frac{60}{y^{5}}$

Step 3

The second derivative of $f (y)$ with respect to $y$ is $- \frac{28}{y^{3}} - \frac{60}{y^{5}}$ .

$- \frac{28}{y^{3}} - \frac{60}{y^{5}}$