Calculus Examples

$J (v) = (v^{3} - 2 v) (v^{- 4} + v^{- 2})$

Step 1

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

$(v^{3} - 2 v) (- 4 v^{- 5} - 2 v^{- 3}) + (v^{- 4} + v^{- 2}) (3 v^{2} - 2 \cdot 1)$

Step 2.8

Multiply $- 2$ by $1$ .

$(v^{3} - 2 v) (- 4 v^{- 5} - 2 v^{- 3}) + (v^{- 4} + v^{- 2}) (3 v^{2} - 2)$

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine terms.

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Step 3.5.1

Combine $- 4$ and $\frac{1}{v^{5}}$ .

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

Step 3.5.2

Move the negative in front of the fraction.

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

Step 3.5.3

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

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

Step 3.5.4

Move the negative in front of the fraction.

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

Step 3.6

Reorder terms.

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

Step 3.7

Simplify each term.

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Step 3.7.1

Expand $(- \frac{4}{v^{5}} - \frac{2}{v^{3}}) (v^{3} - 2 v)$ using the FOIL Method.

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Step 3.7.1.1

Apply the distributive property.

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

Step 3.7.1.2

Apply the distributive property.

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

Step 3.7.1.3

Apply the distributive property.

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

Step 3.7.2

Simplify and combine like terms.

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Step 3.7.2.1

Simplify each term.

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Step 3.7.2.1.1

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

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Step 3.7.2.1.1.1

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

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

Step 3.7.2.1.1.2

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

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

Step 3.7.2.1.1.3

Cancel the common factor.

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

Step 3.7.2.1.1.4

Rewrite the expression.

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

Step 3.7.2.1.2

Move the negative in front of the fraction.

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

Step 3.7.2.1.3

Cancel the common factor of $v$ .

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Step 3.7.2.1.3.1

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

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

Step 3.7.2.1.3.2

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

$- \frac{4}{v^{2}} + \frac{- 4}{v \cdot v^{4}} (- 2 v) - \frac{2}{v^{3}} v^{3} - \frac{2}{v^{3}} (- 2 v) + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.3.3

Factor $v$ out of $- 2 v$ .

$- \frac{4}{v^{2}} + \frac{- 4}{v \cdot v^{4}} (v \cdot - 2) - \frac{2}{v^{3}} v^{3} - \frac{2}{v^{3}} (- 2 v) + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.3.4

Cancel the common factor.

$- \frac{4}{v^{2}} + \frac{- 4}{v \cdot v^{4}} (v \cdot - 2) - \frac{2}{v^{3}} v^{3} - \frac{2}{v^{3}} (- 2 v) + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.3.5

Rewrite the expression.

$- \frac{4}{v^{2}} + \frac{- 4}{v^{4}} \cdot - 2 - \frac{2}{v^{3}} v^{3} - \frac{2}{v^{3}} (- 2 v) + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.4

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

$- \frac{4}{v^{2}} + \frac{- 4 \cdot - 2}{v^{4}} - \frac{2}{v^{3}} v^{3} - \frac{2}{v^{3}} (- 2 v) + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.5

Multiply $- 4$ by $- 2$ .

$- \frac{4}{v^{2}} + \frac{8}{v^{4}} - \frac{2}{v^{3}} v^{3} - \frac{2}{v^{3}} (- 2 v) + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.6

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

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Step 3.7.2.1.6.1

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

$- \frac{4}{v^{2}} + \frac{8}{v^{4}} + \frac{- 2}{v^{3}} v^{3} - \frac{2}{v^{3}} (- 2 v) + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.6.2

Cancel the common factor.

$- \frac{4}{v^{2}} + \frac{8}{v^{4}} + \frac{- 2}{v^{3}} v^{3} - \frac{2}{v^{3}} (- 2 v) + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.6.3

Rewrite the expression.

$- \frac{4}{v^{2}} + \frac{8}{v^{4}} - 2 - \frac{2}{v^{3}} (- 2 v) + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.7

Cancel the common factor of $v$ .

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Step 3.7.2.1.7.1

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

$- \frac{4}{v^{2}} + \frac{8}{v^{4}} - 2 + \frac{- 2}{v^{3}} (- 2 v) + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.7.2

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

$- \frac{4}{v^{2}} + \frac{8}{v^{4}} - 2 + \frac{- 2}{v \cdot v^{2}} (- 2 v) + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.7.3

Factor $v$ out of $- 2 v$ .

$- \frac{4}{v^{2}} + \frac{8}{v^{4}} - 2 + \frac{- 2}{v \cdot v^{2}} (v \cdot - 2) + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.7.4

Cancel the common factor.

$- \frac{4}{v^{2}} + \frac{8}{v^{4}} - 2 + \frac{- 2}{v \cdot v^{2}} (v \cdot - 2) + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.7.5

Rewrite the expression.

$- \frac{4}{v^{2}} + \frac{8}{v^{4}} - 2 + \frac{- 2}{v^{2}} \cdot - 2 + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.8

Combine $\frac{- 2}{v^{2}}$ and $- 2$ .

$- \frac{4}{v^{2}} + \frac{8}{v^{4}} - 2 + \frac{- 2 \cdot - 2}{v^{2}} + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.1.9

Multiply $- 2$ by $- 2$ .

$- \frac{4}{v^{2}} + \frac{8}{v^{4}} - 2 + \frac{4}{v^{2}} + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.2

Add $- \frac{4}{v^{2}}$ and $\frac{4}{v^{2}}$ .

$\frac{8}{v^{4}} - 2 + 0 + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.2.3

Add $\frac{8}{v^{4}}$ and $0$ .

$\frac{8}{v^{4}} - 2 + (3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.3

Expand $(3 v^{2} - 2) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$ using the FOIL Method.

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Step 3.7.3.1

Apply the distributive property.

$\frac{8}{v^{4}} - 2 + 3 v^{2} (\frac{1}{v^{4}} + \frac{1}{v^{2}}) - 2 (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.3.2

Apply the distributive property.

$\frac{8}{v^{4}} - 2 + 3 v^{2} \frac{1}{v^{4}} + 3 v^{2} \frac{1}{v^{2}} - 2 (\frac{1}{v^{4}} + \frac{1}{v^{2}})$

Step 3.7.3.3

Apply the distributive property.

$\frac{8}{v^{4}} - 2 + 3 v^{2} \frac{1}{v^{4}} + 3 v^{2} \frac{1}{v^{2}} - 2 \frac{1}{v^{4}} - 2 \frac{1}{v^{2}}$

Step 3.7.4

Simplify and combine like terms.

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Step 3.7.4.1

Simplify each term.

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Step 3.7.4.1.1

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

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Step 3.7.4.1.1.1

Factor $v^{2}$ out of $3 v^{2}$ .

$\frac{8}{v^{4}} - 2 + v^{2} \cdot 3 \frac{1}{v^{4}} + 3 v^{2} \frac{1}{v^{2}} - 2 \frac{1}{v^{4}} - 2 \frac{1}{v^{2}}$

Step 3.7.4.1.1.2

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

$\frac{8}{v^{4}} - 2 + v^{2} \cdot 3 \frac{1}{v^{2} v^{2}} + 3 v^{2} \frac{1}{v^{2}} - 2 \frac{1}{v^{4}} - 2 \frac{1}{v^{2}}$

Step 3.7.4.1.1.3

Cancel the common factor.

$\frac{8}{v^{4}} - 2 + v^{2} \cdot 3 \frac{1}{v^{2} v^{2}} + 3 v^{2} \frac{1}{v^{2}} - 2 \frac{1}{v^{4}} - 2 \frac{1}{v^{2}}$

Step 3.7.4.1.1.4

Rewrite the expression.

$\frac{8}{v^{4}} - 2 + 3 \frac{1}{v^{2}} + 3 v^{2} \frac{1}{v^{2}} - 2 \frac{1}{v^{4}} - 2 \frac{1}{v^{2}}$

Step 3.7.4.1.2

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

$\frac{8}{v^{4}} - 2 + \frac{3}{v^{2}} + 3 v^{2} \frac{1}{v^{2}} - 2 \frac{1}{v^{4}} - 2 \frac{1}{v^{2}}$

Step 3.7.4.1.3

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

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Step 3.7.4.1.3.1

Factor $v^{2}$ out of $3 v^{2}$ .

$\frac{8}{v^{4}} - 2 + \frac{3}{v^{2}} + v^{2} \cdot 3 \frac{1}{v^{2}} - 2 \frac{1}{v^{4}} - 2 \frac{1}{v^{2}}$

Step 3.7.4.1.3.2

Cancel the common factor.

$\frac{8}{v^{4}} - 2 + \frac{3}{v^{2}} + v^{2} \cdot 3 \frac{1}{v^{2}} - 2 \frac{1}{v^{4}} - 2 \frac{1}{v^{2}}$

Step 3.7.4.1.3.3

Rewrite the expression.

$\frac{8}{v^{4}} - 2 + \frac{3}{v^{2}} + 3 - 2 \frac{1}{v^{4}} - 2 \frac{1}{v^{2}}$

Step 3.7.4.1.4

Combine $- 2$ and $\frac{1}{v^{4}}$ .

$\frac{8}{v^{4}} - 2 + \frac{3}{v^{2}} + 3 + \frac{- 2}{v^{4}} - 2 \frac{1}{v^{2}}$

Step 3.7.4.1.5

Move the negative in front of the fraction.

$\frac{8}{v^{4}} - 2 + \frac{3}{v^{2}} + 3 - \frac{2}{v^{4}} - 2 \frac{1}{v^{2}}$

Step 3.7.4.1.6

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

$\frac{8}{v^{4}} - 2 + \frac{3}{v^{2}} + 3 - \frac{2}{v^{4}} + \frac{- 2}{v^{2}}$

Step 3.7.4.1.7

Move the negative in front of the fraction.

$\frac{8}{v^{4}} - 2 + \frac{3}{v^{2}} + 3 - \frac{2}{v^{4}} - \frac{2}{v^{2}}$

Step 3.7.4.2

Combine the numerators over the common denominator.

$\frac{8}{v^{4}} - 2 + \frac{3 - 2}{v^{2}} + 3 - \frac{2}{v^{4}}$

Step 3.7.4.3

Subtract $2$ from $3$ .

$\frac{8}{v^{4}} - 2 + \frac{1}{v^{2}} + 3 - \frac{2}{v^{4}}$

Step 3.8

Combine the numerators over the common denominator.

$- 2 + 3 + \frac{8 - 2}{v^{4}} + \frac{1}{v^{2}}$

Step 3.9

Subtract $2$ from $8$ .

$- 2 + 3 + \frac{6}{v^{4}} + \frac{1}{v^{2}}$

Step 3.10

Add $- 2$ and $3$ .

$1 + \frac{6}{v^{4}} + \frac{1}{v^{2}}$